The Multiple Zeta Value Data Mine

DESY 09-003 arXiv:0907 2557; [math-ph] NIKHEF 09-016 SFB/CPP -09-65 The Multiple Zeta V alue Data Mine J. Bl ¨ umlein a , D.J. Broadhurst b , J.A.M. V ermaseren a , c 1 a Deutsches Elektr onen-Sync hr otr on, DESY , Platanenallee 6 , D-15738 Zeuthen, Germany b Physics and Astr o n omy Department, Open Uni versity , Milton Ke ynes MK7 6AA, UK c Nikhef Theory Gr oup Science P ark 10 5, 109 8 XG Amster dam, The Netherlands Abstract W e pro vide a data mine of prove n results for multipl e zeta valu es (MZVs) of the form ζ ( s 1 , s 2 , . . . , s k ) = ∑ ∞ n 1 > n 2 >...> n k > 0  1 / ( n s 1 1 ... n s k k )  with weight w = ∑ k i = 1 s i and depth k and for Euler sums of the form ∑ ∞ n 1 > n 2 >...> n k > 0  ( ε n 1 1 ... ε n k 1 ) / ( n s 1 1 ... n s k k )  with signs ε i = ± 1. Notably , we achie ve explici t prove n reductions of all MZ Vs with weights w ≤ 22, and all Euler sums with weights w ≤ 12, to b ases whose dimensio ns, bigrad ed by weight and depth, hav e sizes in precise agreement with the Broadhu rst– Kreimer and Broadhurst con jecture s. Moreo ver , we lend furth er supp ort to these conjec tures by studying e ven greate r weights ( w ≤ 30), using modul ar ari thmetic. T o obtain these resul ts we deri ve a n e w type of re lation for Euler sums, the Gene ralized Doubling R elation s. W e elucidate the “pushdo wn” mechani sm, whereby the ornate enumerat ion of primiti ve MZVs, by weight and depth, is reconciled with the far simpler en umeratio n of primiti ve Euler s ums. There is so me e vidence that thi s push- do w n mechanism ﬁnds its origin in doubling relations. W e hope that our data mine, obtain ed by ex ploitin g the uniqu e power of the computer algebra language F O R M , will ena ble the stud y of many more such conseq uence s of the double -shuf ﬂe algebra of MZ Vs, and their E uler cousins, which are already the subje ct of keen intere st, to practit ioners of quantum ﬁ eld theo ry , and to mathematicians alike. 1 Alexander-von-Humboldt A wardee. 1 Introduction Multiple Zeta V alues (MZVs) and Euler sums [1–3] ha ve been of interest to m athemati- cians [1, 4–7] and phys icists [8] for a long t ime. One place in p hysics in which they are important is perturbative Quantum Field Theory . The interest became even larger when higher order calculat ions i n Quantum Electrodynam ics (QED) and Quant um Chromody- namics (QCD) started to need t he mult iple harmonic sum s S ~ c ( N ) [9–11]. Eul er sums are obtained as the limit N → ∞ of the related multi ple sums Z ~ c ( N ) ζ ~ c = ∞ ∑ k = 1 ( σ ( b )) k k | b | Z ~ a ( k − 1 ) , (1.1) with ~ c = ( b ,~ a ) , b , a i ∈ Z and Z b , ~ a ( N ) = N ∑ k = 1 ( σ ( b )) k k | b | Z ~ a ( k − 1 ) , Z / 0 = 1 , Z ~ a ( 0 ) = 0 , (1.2) with σ ( b ) = s ign ( b ) . Euler sums for which all indices are positive are called Multipl e Zeta V alues. Euler sums and MZVs with the ﬁrst index b = 1 dive r ge, but wi ll be in- cluded symbolically in the foll owing, for con venience. Their degre e of di verge nce can be uniquely traced back t o a poly nomial in the single harmon ic sum S 1 ( ∞ ) = ∑ N → ∞ ∑ N k = 1 1 k shown later in the text. W e call the num ber of indices of the Euler sum s and M ZVs their depth d and w = d ∑ k = 1 | c k | (1.3) their weight. The number of Euler sums , resp. MZVs, up t o a give n weight w grows rapidly and amounts t o 2 · 3 w − 1 and 2 w − 1 , respecti vely . A central question thus concerns to ﬁnd all the relations between the Euler sums, resp. MZVs for ﬁxed weight and depth , and e ven more im portantly , new r elat ions between MZVs at the one hand and Euler sums on the other hand, and the corresponding bases. Besides weight and depth, another degree of freedom, being discuss ed later , the pushdown p , quantiﬁes the relatio n between MZVs and Euler su ms. The way to view MZVs, embedded into E uler sums, dates back to Broadhurst [12], who conjectured the counti ng of basis elements at ﬁxed { w , d } . Th e corresponding conjecture for the MZVs is due to Broadhurst and Kreimer [13] 2 . For the number of basi s elements for MZVs of a given weight, without regard to depth, an upper bound has been proven in [14]. This coincides with the result obtained by sum ming t he numbers conjectured in [13] over all d epths at a ﬁxed wei ght. The relations between MZVs and Euler sums in Ref. [12] are conjectured using algo- rithms for integer relatio ns as PSLQ [15] and LLL [16] wh ich use representations based on a large number of digits. It is well-known that M ZVs obey shuf ﬂe- and stuf ﬂe-relations. This is due to their representation i n term s of Poincar ´ e iterated integrals [17] at argument x = 1, whi ch are 2 Conjectures f or ﬁxed weig ht are due to Zagier [2] and pro bably also indepen dently due to Drinfel’d, Gonchar ov an d K o ntsevich. 1 harmonic polyl ogarithms [18] on the on e hand, and harmonic sum s [9–11] on the other hand. The former qu antities obey a shufﬂe - the latter a quasi-shufﬂe algebra, i.e. shuf- ﬂing wit h “stuff ” from polynom ials of harmonic s ums of lower weight. Currently no other independent relation is kno wn between M ZVs. The Eu ler s ums are also related by both the shuf ﬂe- and stufﬂe-relations, where now also negative indices occur to indi cate alternating sum s. Howe ver , these relation s are not su f ﬁcient t o obtain the minimal s et of basis elements as bein g conjectured in [12]. Starting with w = 8 it requires the do ubling relation and with w = 1 1 generalized dou bling relations deriv ed in the present paper . Be- ginning wi th w = 12 relations o ccur , which allow to express MZVs of a given depth in terms of Euler sums of a lesser depth. Part of these relations ha ve been conjectured in the past using integer relations [12, 19]. A main objective of th e present paper is to prove these relati ons applying computer algebra methods and to ﬁnd relations of this ty pe in a more systematic way . W e in vestigate the Euler su ms to w = 12 compl etely , deri ving basis-representations for all i ndividual v alues in an explicit analytic calculation . For the MZVs th e sam e anal- ysis i s being performed up to w = 22 . T o w = 24 we checked the conj ectured si ze of the basis usi ng modul ar arithm etic. Under the further conjecture that the basis elements can be chosen out of M ZVs of depth d ≤ w / 3 we conﬁrm the conjecture up to w = 26 . Fur- thermore, the following runs at l imited depth, usi ng m odular arithmetic keeping the hi gh- est weig ht terms only , were performed: d = 7 , w = 27 ; d = 6 , w = 28 ; d = 7 , w = 2 9 ; d = 6 , w = 30 . For the Eu ler sums comp lete result s were obtained for d ≤ 3 , w = 29 ; d ≤ 4 , w = 22 ; d ≤ 5 , w = 17 and for d ≤ 3 , w = 51 ; d ≤ 4 , w = 30 ; d ≤ 5 , w = 21 ; d ≤ 6 , w = 17 using modular arithmetic n eglecting prod ucts o f lo wer weight. The con- jectures on the n umber of basis elements w .r .t. { w , d } were veriﬁed in all t hese cases. The results of our analysi s are made ava ilable in the Multip le Zeta Data Mine [20], to allow users to search for yet un-discovered relati ons. The paper i s organized as follows. In Section 2 we summarize basic n otations and the well k nown relations between Eu ler s ums. A nove l typ e of relations, the generalized doubling relations, is deri ved in Section 4. Th ere we als o discuss i ts impact i n ﬁ nding the basis elements at a given weight w and depth d . In Section 5 an outl ine is given on the details of the computer algebra code, which allowed to deri ve the basis-representations of th e MZVs and Euler sums. Details on th e runnin g for the different cases are reported in Section 6 . The results are stored in the Multiple Zeta V alue Data Mine 3 , which is described in Section 7. T o establi sh the solution of the problem s dealt with in the current project required som e new features of FORM [21] and TFORM [22], whi ch are described in Section 8. In Section 9 we brieﬂy revie w the status achieved by other groups and present ﬁrst results of the analysis. In particular a series of conjectures made in the mathematical literature are conﬁrmed within t he range explored in the present s tudy . Here we discuss also particular choices for the respectiv e bases. An int eresting aspect representing MZVs by Euler sums concerns the so-called pushdowns, i.e. the representation o f a MZV of a giv en depth d with Euler sums of dept h d ′ with d ′ < d . These are studied in Section 1 0 in whi ch w e als o introd uce a ne w kind of object, the A ~ a –functions. The y p lay a key role in representing a class of Euler sums. Some mo re special Euler su ms are studied i n 3 It goes without saying that also the Euler sums are covered here. 2 Section 11. Section 12 contains the conclusions and an outl ook. In the Appendices we provide dif ferent basis representations and discuss the pushd owns in more detail. 2 Basic F orm alism In the following we work with three types of objects, the ﬁnite nested harmonic S ~ a -sums, Z ~ a -sums, both at argument N ∈ N , and the harmonic pol ylogarithms H ~ a at ar gument x , 0 ≤ x ≤ 1. The y all can b e used to deﬁne the MZVs and the Euler su ms in th e limit N → ∞ and x = 1 , respectively . W e generally consider the case of colored obj ects corresponding to n = 2, i.e. numerator weights with ( ± 1 ) k , i.e. polylogarit hms of square root of unity . The harmonic S -sums are deﬁned by S ~ a ( 0 ) = 0 S b ( N ) = N ∑ k = 1 ( σ ( b )) k k | b | S b , ~ a ( N ) = N ∑ k = 1 ( σ ( b )) k k | b | S ~ a ( k ) . (2.1) In this form these sums are usually us ed by physicis ts. In particular result s in QCD [23–26] are expressed in terms of these objects 4 . Next there are the Z -su ms. They are deﬁ ned in (1. 2). These are of c ourse very s imilar to the S -sums and it is straightforward t o con vert from one notati on to the ot her . The Z -sums are mostly used by mathemati cians. In the limit N → ∞ and when σ ( b ) = 1 for all b they deﬁne the Multiple Zeta V alu es (MZVs): ζ ~ a = lim N → ∞ Z ~ a ( N ) . (2.2) When we allow σ ( b ) to take the values + 1 or − 1 and we take the limi t N → ∞ we speak of Euler sums. Finally , there are the harmonic polylogarithm s, which we will also call H -functions. W e consider the alphabets h = { 0 , 1 , − 1 } and H = { 1 / x , 1 / ( 1 − x ) , 1 / ( 1 + x ) } , (2.3) which deﬁne the elements of the inde x set of t he harmonic polylogarithm s 5 and the func- tions in the iterated in tegrals, respectively . Let ~ a = { m 1 , . . . , m k } , m i , b ∈ h , k ≥ 1, 4 The class of Euler sums is known to be too small in general to represent all Feynman diagrams f or no-scale processes in scalar ﬁeld theories, b ut have to be e xtended in higher orders [2 7–30]. This will app ly also for ﬁeld theories as QCD and QED. Feynman-integrals are p eriods [ 31] if all ratios of Lorenz invariants and masses have rational values [32 ]. 5 Special c ases are the classical poly logarithm s [33 ] and the Nielsen polylogar ithms [ 34]. Gen eraliza- tions of harmon ic polylogarithms are fou nd in [35, 3 6]. 3 then H b , ~ a ( x ) = Z x 0 d z f b ( z ) H ~ a ( z ) f 0 ( z ) = 1 / z f 1 ( z ) = 1 / ( 1 − z ) f − 1 ( z ) = 1 / ( 1 + z ) H 0 ( x ) = log ( x ) H 1 ( x ) = − log ( 1 − x ) H − 1 ( x ) = log ( 1 + x ) . (2.4) The sum s to in ﬁnity and the H -functions at unity are all related and can be readily trans- formed int o each other . For some applicatio ns it is most con venient to work wi th one set of obj ects and for ot hers other objects m ay b e more useful. For reasons being explained later our computer programs work mostly with H -functions at unity . A ﬁrst aspect to note is that t he index ﬁelds of the sum s and the functions are of a diffe rent nature. This can be seen b y introducing t he notation in w hich the index n in the sums can alternativ ely be writ ten as n − 1 zeroes fol lowed by a one and − n is wri tten as n − 1 zeroes followed b y a minus one. In the H -funct ions we can absorb alternati vely t he zeroes in the no nzero number to their right by raising it s abso lute value by one for each zero being absorbed. This leaves only the ri ghtmost zeroes. Hence: S − 3 , 4 ( N ) = S 0 , 0 , − 1 , 0 , 0 , 0 , 1 ( N ) Z 2 , − 5 ( N ) = Z 0 , 1 , 0 , 0 , 0 , 0 , − 1 ( N ) H 0 , 1 , − 1 , 0 , 0 , − 1 , 0 , 0 ( x ) = H 2 , − 1 , − 3 , 0 , 0 ( x ) . (2.5) The notation in terms of the 0 , ± 1 we call the (iterated) integral notation. The natural notation of the sums we call the (nested) sum notation. Reference to the alphabet h allows us to count the number of objects and to classify them. T he number of indices in this int egral no tation is called the weight of the s um or the function. For a given wei ght w there are 2 · 3 w − 1 sums and 3 w H -function s. When the sums are written i n the original sum not ation, the number of indices indicates the number of nested sums. This is also called the depth o f the sum. When th ere are no trailing zeroes in the H -functions we can introduce the depth in the same way . Because of algebraic relations we can express the functions with trailing zeroes as products of po wers of log ( x ) and H -functions with fewer indi ces [18, 37]. In t hat case the concept of depth can be used in a simil ar way as with the sum s. For an y ar gument x 6 = 1 the H -functions form a sh uf ﬂe algebra: H ~ p ( x ) H ~ q ( x ) = ∑ ~ r ∈ ~ p ⊔ ⊔ ~ q H ~ r ( x ) , (2.6) where ~ p ⊔ ⊔ ~ q denotes the s hufﬂ e product, cf. e.g. [37], and p i , q i ∈ h . When x = 1 H - functions for which th e ﬁrst index is one are div er gent. It i s howe ver po ssible t o express them in terms of a s ingle div er g ent object and other ﬁnite terms in a consistent way . The only thin g that breaks d own is that there are correction terms to the sh ufﬂ e relations 4 when both objects in t he l eft hand side are diver gent , see also Ref. [18]. Because t he number of non-zero indi ces remai ns the same during the shufﬂe operation, we call it depth preserving. For general argument N the sums form a stu f ﬂe algebra, [37]. This is a general prop- erty of sums which we show here for a double sum: N ∑ i = 1 N ∑ j = 1 = N ∑ i = 1 i ∑ j = 1 + N ∑ j = 1 j ∑ i = 1 − N ∑ i = j = 1 = N ∑ i = 1 i − 1 ∑ j = 1 + N ∑ j = 1 j − 1 ∑ i = 1 + N ∑ i = j = 1 . (2.7) The diagonal terms give extra ‘stuff ’ beyond the no rmal shufﬂing in t he n atural notation for the sum s. Even though the diagonal terms add terms usually the st uf ﬂe relations ha ve fe wer terms because most of the time s ome of th e i ndices will have an absolute va lue greater than one. W e write in terms of S – or Z –notation : S m ( N ) S n ( N ) = S m , n ( N ) + S n , m ( N ) − S m & n ( N ) (2.8) S m ( N ) S n , k ( N ) = S m , n , k ( N ) + S n , m , k ( N ) + S n , k , m ( N ) − S m & n , k ( N ) − S n , m & k ( N ) (2.9) Z m ( N ) Z n ( N ) = Z m , n ( N ) + Z n , m ( N ) + Z m & n ( N ) (2.10) Z m ( N ) Z n , k ( N ) = Z m , n , k ( N ) + Z n , m , k ( N ) + Z n , k , m ( N ) + Z m & n , k ( N ) + Z n , m & k ( N ) . (2.11) Here the operator & is deﬁned by m & n = σ ( m ) σ ( n )( | m | + | n | ) = σ n m + σ m n . (2.12) The abov e algebraic relations can be used to b ring an expression with many harmonic polylogarithm s or harmo nic sums into a standard form. For ev aluation, h owe ver , it is often useful to work it the other way and reduce t he numb er of objects at the highest weight in fa vor of products of obj ects with a lower weight which are easier to ev aluate. For this th e theory of L yndon w ords [38] applies, b ut especially with the stufﬂe s the e xtra terms which hav e the same weight but a l ower depth have to be taken along and make things considerably more in volved than p ure shufﬂ es. A k -ary L yndon word of l ength n is a n -lett er concatenation product over an alph abet of si ze k , which, observing lexicographical ordering is smaller than all its suf ﬁxes. Equiv- alently , it is t he unique mi nimal element in the lexicographical ordering of all its c yclic permutations. The uniqueness impli es that a L yndon word is ape riodic. So it differs from any of its non-trivial rotations . In our case we will usually replace min imal by maxim al when we form L ynd on words of indices of MZVs or Euler sums. That is, we will put th e lar ger indices to the left. One could also say that the concept of greater than is deﬁned in a special way inside the alphabet. The practical adv antage is that this guarantees that none of the MZVs of which the index string forms a L yndon word is di ver gent. 5 When we use t he s tufﬂ e relations to simpl ify the set of objects at a giv en weigh t, we can arrange that t hey are us ed in such a way that they neve r raise th e v al ue of t he depth parameter . Some terms will h a ve a lower v alue for the depth. Therefore we call the stufﬂes po tentially depth lowering. When we consider the sums to inﬁnity there are two classes of extra relations worth mentioning . The ﬁrst is the ‘rule of the triangle’ which is based on lim N → ∞ N ∑ i = 1 N ∑ j = 1 = lim N → ∞ N ∑ i = 1 N − i ∑ j = 1 + lim N → ∞ N ∑ i = 1 N ∑ j = N − i + 1 . (2.13) For most sums the second term will give a limit that goes t o zero wi th at least one power of 1 / N , poss ibly m ultipli ed by power s of log ( N ) . Thi s system can be generalized to the product of any pair of sums and it can be proven that the limit of the second term v anishes when at least o ne of the sums in the left hand si de is ﬁnite [10]. When both are diver g ent it is possi ble t o work out which e xtra terms are n eeded. Because the sums of the ﬁ rst term in t he right hand side c an be worked out, e ven in the most general case, th e above giv es us an extra algebraic relation for the sums to inﬁnity . These relations are depth preserving. When we consider the H -functions at unity , it is easy to see that the y can be writ ten as nested sums t o inﬁnity of the same variety as the Z -sums or the S -sums. Hence th ey no w obey also the stufﬂe algebra. And it can be shown that the ‘rule of th e triangl e’ is no m ore than the equiv alent of the shufﬂe algebra for the H -functi ons, with the same restrictions for the double div er gent terms. The next set of relations is easy to see for ﬁnite sums: S m ( N ) = N ∑ i = 1 1 i m = N ∑ i = 1 2 m 1 ( 2 i ) m = 2 N ∑ i = 1 2 m − 1 1 + ( − 1 ) i i m = 2 m − 1 [ S m ( 2 N ) + S − m ( 2 N )] , (2.14) which generalizes into S n 1 , ··· , n p ( N ) = 2 n 1 + ··· + n p − p ∑ ± S ± n 1 , ··· , ± n p ( 2 N ) . (2.15) Here the sum is over all 2 p plus/min us com binations. These relations are called the ‘dou- bling relations’. For ﬁnite sums with n 1 6 = 1 these relations can be used directly . In the case that dive r gent sums are in volved there are again correction terms. The equiv alent form ula for t he H -functions is obtained by lo oking at H ~ a ( x 2 ) and noti c- ing that at x = 1 this is the same as H ~ a ( x ) . In that case we have H 1 , 0 , 1 ( x 2 ) = 2 [ H 1 , 0 , 1 ( x ) − H − 1 , 0 , 1 ( x ) − H 1 , 0 , − 1 ( x ) + H − 1 , 0 , − 1 ( x )] , (2.16) which generalizes t o any num ber of indices. The rule is t hat the fac tor is i dentical to 2 m in which m is the number o f zeroes i n the i ndices, and each one in the left hand side gi ves a doubli ng of terms in the right hand sid e: one term with a corresponding 1 and one with a corresponding − 1 and an extra overall minu s si gn. In the left hand side one cannot ha ve negati ve indi ces. Again one should be careful with the di ver gent functions. 6 Div er g ences are e xpressed in t erms of the ob ject S 1 ( ∞ ) . In mo st cases one can use this as a regular sym bol and take it along in the equations and e xpressi ons. Unless we mention the prob lems explicitl y , one can exchange li mits and sums when this object i s combined with ﬁnite s ums. The reason is that our ﬁnite sums con ver ge faster t han that th is object div er ges. A problem occurs when we use the doubling formula on it. W e ﬁnd: S 1 ( ∞ ) = S 1 ( 2 ∞ ) + S − 1 ( 2 ∞ ) = S 1 ( 2 ∞ ) − log ( 2 ) , (2.17) which just shows that the di ver gence of S 1 ( ∞ ) is log arithmic, since S 1 ( N ) = ln ( N ) + γ E + 1 2 N + 1 12 N 2 + O  1 N 3  , (2.18) cf. [25]. One can howe ver use the st ufﬂ e relations on these objects. This allows one in principle to expre ss the diver gent sums in terms of products of S 1 ( ∞ ) and ﬁnite sum s as in S 1 ( N ) S m , n ( N ) = S 1 , m , n ( N ) + S m , 1 , n ( N ) + S m , 1 , n ( N ) − S m &1 , n ( N ) − S m , n &1 ( N ) . (2.19) If we assume m 6 = 1 thi s allows us to express the diver gent sum S 1 , m , n ( ∞ ) the way we want i t. Similarly one ca n no w look at stufﬂes of S 1 · S 1 to determine S 1 , 1 and then look at stufﬂes of S 1 , 1 ( N ) wi th ﬁnit e sums. In the prog rams we give S 1 ( ∞ ) the nam e Sinf which, due to the above, can be treated as a regular symbol. Because we have two shufﬂe produ cts - the stufﬂe-algebra i s a quasi-shufﬂe algebra [39] - we can equate the result of the stufﬂe product of two objects wit h the result of the shufﬂe product of t he sam e t wo objects. Th e resulti ng relation is called a dou ble-shufﬂe relation and con tains only objects of the same weight . Th ese relations have been used in a number o f calculati ons. For our type of calculations they are, ho we ver , not s uitable. W e will use the stufﬂe and the shufﬂe relations i ndividually . This will allow a bett er optimizatio n of the al gorithms. The concept of dua lity is very useful and allows us to roughly half the nu mber of objects that need to be com puted. Th e dualit y relati on is deﬁned in the integral not ation using harmonic pol ylogarithms a t one. It states that if we have a MZV and we re verse the order of its indices while at the same time transforming zeroes into ones and on es into zeroes the new object has the same v alue a s the origin al. An example of this duality is the relation H 0 , 1 , 0 , 1 , 1 , 1 , 1 , 1 = H 0 , 0 , 0 , 0 , 0 , 1 , 0 , 1 (2.20) In mathematics one traditionally considers this duali ty in su m notation. In that ca se, for a sequence I = ( p 1 + 1 , { 1 } q 1 − 1 , p 2 + 1 , { 1 } q 2 − 1 , . . . , p k + 1 , { 1 } q k − 1 ) (2.21) there is a dual sequence τ ( I ) = ( q k + 1 , { 1 } p k − 1 , q k − 1 + 1 , { 1 } p k − 1 − 1 , . . . , q 1 + 1 , { 1 } p 1 − 1 ) . (2.22) 7 The duality theorem [2] states ζ I = ζ τ ( I ) . (2.23) It was conjectured in [40] and is easily proven by the transformation x → 1 − t of the corresponding iterated integrals. Because for e ven weights there are some elements t hat are self-dual this does not divide the n umber of terms exactly b y two. Considering that we do not ha ve to consi der the dive r gent objects we hav e 2 w − 3 rele va nt objects when w i s odd and 2 w − 3 + 2 w / 2 − 2 rele va nt objects when w is ev en. For Euler sums the equivalent transformation is more complicated due to the th ree letter alphabet. It is obtain ed by studying the transformation x → 1 − t 1 + t (2.24) in the integral representation. Its ef fect is that gi ven the alph abet A = 0 ← 1 x B = 1 ← 1 1 − x C = − 1 ← 1 1 + x (2.25) and a string of letters from t his alphabet as i ndices of an E uler sum H , the ‘dual expres- sion’ is obtained by re verting the string of letters and making the replacement A → B ⊕ C B → A ⊖ C C → C . (2.26) The addition and subtraction operators here mean th at for each such transformation there will b e a doubling of the number of terms , one wit h the ﬁrst letter and the other wit h the other letter . Th e sign-operator ⊕ ( ⊖ ) refers to the sig n of the complete term. Be cause these relations can both raise and lower t he depth of a term we call them depth mixing. W e have tested th at this transform ation does add something new b eyond what the stufﬂes and the shufﬂes giv e us. In particular , when one derives equations for all sums at a given weight , they can be used to replace the doubling and the Generalized Dou b ling Relations (GDRs), see Section 4. W e ha ve t ested th is to weight w = 12 . Unfortunately they cannot be used when the concept of depth of the sums is important and hence we hav e no t used these equations in our programs. A generalization of the Riemann ζ -function is Hurwitz’ ζ -function [6, 41] : ζ ( n , a ) = ∞ ∑ k = 1 ( sign ( n )) k ( k + a ) | n | , (2.27) which can be e x tended to gener alized Euler sums analogous to (1.1 ). Since a is a real pa- rameter , one may differentiate ζ ( ~ c , a ) w .r .t. a and seek for ne w relations. W e in vestigated this possibil ity , b ut did not ﬁnd new relations beyond thos e quoted above . 8 When we are discuss ing bases into which we write the MZVs and th e Euler sums we recognize two types of basis : Deﬁnition. A basis of a vector space of all Eul er sums or M ZVs at a given wei ght w is called a F ibonacci basi s . Deﬁnition. A basis o f the ring of all Euler sums or MZVs at a given weight w is called a L yndon basis if all its elements ha ve an index ﬁe ld that forms a L yndon word. In a Fibonacci basis all basis elements are nested sums of t he same weight. The name deriv es from th e o bserva tion that the size of such a basis for the Eu ler s ums seem to follow a Fibonacci rule [42]. Also the MZVs seem to follow the rule that the total number of their b asis el ements follo w the Fibonacci-like Padov an num bers [ 43], see App endix A. In a L yndon basis we write in the complete basis as many elem ents as po ssible as products o f lower weight b asis element s and what remai ns is the L yndo n basis. Simulta- neously we require the in dex ﬁeld to form a L ynd on word. Sometimes a L yndon basi s can be formed from a Fibonacci basis by just selecting the L yn don words from i t. The number of basis elements in the case of MZVs is counted by a W itt-type relation [44] based on the Perrin numbers [45]. In th e case of the Euler sums the corresponding relation relies on the Lucas numbers [46], see Appendix A. Any other basis we will call a mixed basis. W e wi ll usually try to arrange the L yn don bases in such a way t hat they are ‘mini- mal depth’. T his means that if an element can be expressed in terms of objects with a lower depth, i t cannot be a member of the basis. Details on a variety of bases are g iv en in Appendix A. The complete basis we actually selected for the M ZVs is presented in Appendix B. 3 Conjectu res on Bases at Fixed W eight and Depth Broadhurst [12] and Broadhurst and Kreimer [13] formulated conjectures on the size of the basis for Euler sums and MZVs, respectiv ely , which we summarize in the follo wi ng. Euler sums ζ ~ a at gi ven weight and depth w , d are called ind ependent if there exists no relation between them, cf. Sect. 2,4. The elem ents of the b asis through which all Euler sum s can be represented in t erms of polynomials are called pr imitive . The num- bers of independent and prim itive sums at a giv en weight are ﬁxed, while di f ferent basi s representations may be chosen. Let E w , d be the number of independent Euler sums at weight w > 2 and dept h d that cannot be reduced to primitive Eul er s ums of lesser depth and their products. Thus we believ e that E 3 , 1 = 1, since th ere is n o known relationship between ζ 3 , π 2 and ln ( 2 ) . It is rather natural to guess that E w , d is gi ven by a ﬁltration of the coef ﬁcients of powe rs of x and y in the expansion of 1 / ( 1 − xy − x 2 ) , i.e. that ∏ w > 2 ∏ d > 0 ( 1 − x w y d ) E w , d ? = 1 − xy − x 2 ( 1 − xy )( 1 − x 2 ) = 1 − x 3 y ( 1 − xy )( 1 − x 2 ) . (3.1) It is then easy to obtain E w , d by M ¨ obius transformation of the binomial coefﬁcients in 9 Pascal’ s triangle. Let T ( a , b ) = 1 a + b ∑ d | a , b µ ( d ) ( a / d + b / d ) ! ( a / d ) ! ( b / d ) ! (3.2) where the sum is over all posi tive integers d that divide both a and b and the M ¨ obius function is deﬁned by µ ( d ) =    1 when d = 1 0 when d is divisible by the square of a prime ( − 1 ) k when d is the product of k distinct primes (3.3) When w and d h a ve the same parity , and w > d , one obt ains from (3.1) E w , d = T  w − d 2 , d  . (3.4) W ith the exception of l n ( 2 ) and ζ 2 , which act as the seeds xy and x 2 , all element s of the basis are thereby conjecturally enumerated. In this paper we provide extensive e vi dence to support conjecture (3.1). Now let D w , d be the num ber of ind ependent MZVs at w eight w > 2 and depth d that cannot be reduced to pri mitive M ZVs of lesser depth and their products. Thus we believ e that D 8 , 2 = 1 , since there is no kno w n relation ship between the doub le sum Z 5 , 3 = ∑ m > n > 0 1 / ( m 5 n 3 ) and single s ums or t heir produ cts. It is tempting to guess t o that D w , d is generated by ﬁltration of the expansion of 1 / ( 1 − x 2 − x 3 y ) , seeded by π 2 and ζ 3 . But this is not t he case, s ince the solution of the doubl e-shufﬂ e algebra at weight w = 12 lea ves one quadruple sum undetermined, while the obvious guess would lea ve none. The conjecture [13] in this case is rather ornate, cf. T able 16. ∏ w > 2 ∏ d > 0 ( 1 − x w y d ) D w , d ? = 1 − x 3 y 1 − x 2 + x 12 y 2 ( 1 − y 2 ) ( 1 − x 4 )( 1 − x 6 ) (3.5) with a correction term whose numerator , x 12 y 2 ( 1 − y 2 ) , ensu res that D 12 , 4 = 1 and D 12 , 2 = 1, in agreement with the solution of the double-shufﬂ e alg ebra. The denominato r ( 1 − x 4 )( 1 − x 6 ) is t hen chosen to g iv e D 2 m , 2 = ⌊ ( m − 1 ) / 3 ⌋ for the number of primiti ve double sums with weight 2 m . Conj ecture (3.5) is impressive ly s upported by the data mine. Furthermore, ∏ w > 2 ∏ d > 0 ( 1 − x w y d ) M w , d = 1 − x 2 − x 3 y 1 − x 2 (3.6) is the conjectured generating function of the basis elements M w , d of the MZVs when expressed as Euler sums in a minimal depth representation, see T able 17. 4 Generaliz ed Doubli n g Relatio n s Up to w = 10 the shufﬂe -, stufﬂe -, and doubling relatio ns were su f ﬁcient to e xpress the alternating E uler sums over a basis whose si ze is in accordance with the conjecture in 10 Ref. [12]. This is not the case from w = 11 onwards. Th erefore one has to seek a new kind of relations, which we derive in the following. Of course, when we derive all relations at a gi ven w eight we could use the relations of (2.2 6). The fact that they are depth mixing makes t hem u seless for calculations in which the concept of dept h plays a role. Hence we need our new (depth l ower ing) relati ons anyway . W e ﬁrst present the deriv ation of this class of relations which we call Generalized Doubling Relations (abbreviated to GDRs) and di scuss then their effe ct on the number of basis e lements representing the Euler sums. 4.1 Deriv ation of the generalized doubling relations The only relations we could ﬁnd thus far addi ng som ething n e w to the syst em are the depth 2 relati ons o f Ref. [12]. They are based on partial fractioning in two dif ferent w ays. One way is: 1 ( 2 i + j ) ( j ) = 1 ( 2 i + 2 j )( 2 i + j ) + 1 ( 2 i + 2 j )( j ) (4.1) W e can t ake out the fa ctor two and in the ﬁrst term the 2 i is taken care of by changing the summati on o ver i into a sum mation ov er the e ven numbers by including a factor ( 1 + ( − 1 ) i ) / 2, which introd uces negativ e indi ces in some Eul er sums . In the other way we use the more regular form 1 ( 2 i + j ) ( j ) = 1 ( 2 i )( j ) − 1 ( 2 i )( 2 i + j ) . (4.2) T ogether these partial fractions produce new types of relations. Here we will give the new set of relations and their deri vation. W e will work with the Z -sums. The reason is a particularly handy representation of these sums to inﬁni ty [40, 47]: Z m 1 , ··· , m p ( ∞ ) = ∞ ∑ i 1 > i 2 > ··· > i p > 0 σ i 1 1 σ i 2 2 · · · σ i p p i n 1 1 i n 2 2 · · · i n p p = ∞ ∑ x 1 = 1 ∞ ∑ x 2 = 1 · · · ∞ ∑ x p = 1 σ x 1 + x 2 + ··· + x p 1 σ x 2 + ··· + x p 2 · · · σ x p p ( x 1 + x 2 + · · · + x p ) n 1 ( x 2 + · · · + x p ) n 2 · · · ( x p ) n p , (4.3) in which we take n i = | m i | and σ i to be the sign of m i . Let us start with the re-deri vation of the equation for depth d = 2 . Actually we do not reproduce it exactly , b u t we obtain a similar equation. Here we write for bre vity Z ( a , b ) = Z a , b ( ∞ ) . Throu ghout this Section we assume that a , b , c and d are positive integers. W e consider the following com bination of Z -sums : 11 E ( a , b ) = 1 2 ( Z ( a , b ) + Z ( − a , − b )) = ∞ ∑ x 1 = 1 ∞ ∑ x 2 = 1 1 ( x 1 + x 2 ) a x b 2 1 + ( − 1 ) x 1 2 = ∞ ∑ x 1 = 1 ∞ ∑ x 2 = 1 1 ( 2 x 1 + x 2 ) a x b 2 = ∞ ∑ x 1 = 1 ∞ ∑ x 2 = 1 " a ∑ i = 1 A ( a , b ) i 1 ( 2 x 1 + 2 x 2 ) a + b − i ( 2 x 1 + x 2 ) i + b ∑ i = 1 B ( a , b ) i 1 ( 2 x 1 + 2 x 2 ) a + b − i x i 2 # = a ∑ i = 1 A ( a , b ) i 2 i − a − b ∞ ∑ x 1 = 1 ∞ ∑ x 2 = 1 1 ( x 1 + x 2 ) a + b − i ( 2 x 1 + x 2 ) i + b ∑ i = 1 B ( a , b ) i 2 i − a − b Z ( a + b − i , i ) = b ∑ i = 1 B ( a , b ) i 2 i − a − b Z ( a + b − i , i ) + a ∑ i = 1 A ( a , b ) i 2 i − a − b ∞ ∑ x 1 = 1 ∞ ∑ x 2 = x 1 + 1 1 ( x 1 + x 2 ) i x a + b − i 2 = b ∑ i = 1 B ( a , b ) i 2 i − a − b Z ( a + b − i , i ) + a ∑ i = 1 A ( a , b ) i 2 i − a − b ∞ ∑ x 1 = 1 ∞ ∑ x 2 = 1 1 ( x 1 + x 2 ) i x a + b − i 2 − a ∑ i = 1 A ( a , b ) i 2 i − a − b ∞ ∑ x 1 = 1 x 1 ∑ x 2 = 1 1 ( x 1 + x 2 ) i x a + b − i 2 = b ∑ i = 1 B ( a , b ) i 2 i − a − b Z ( a + b − i , i ) + a ∑ i = 1 A ( a , b ) i 2 i − a − b Z ( i , a + b − i ) − a ∑ i = 1 A ( a , b ) i 2 i − a − b ∞ ∑ x 2 = 1 ∞ ∑ x 1 = x 2 1 ( x 1 + x 2 ) i x a + b − i 2 = b ∑ i = 1 B ( a , b ) i 2 i − a − b Z ( a + b − i , i ) + a ∑ i = 1 A ( a , b ) i 2 i − a − b Z ( i , a + b − i ) − a ∑ i = 1 A ( a , b ) i 2 i − a − b ∞ ∑ x 2 = 1 ∞ ∑ x 1 = 1 1 ( x 1 + 2 x 2 ) i x a + b − i 2 − a ∑ i = 1 A ( a , b ) i 2 i − a − b ∞ ∑ x 2 = 1 1 ( 2 x 2 ) i x a + b − i 2 = b ∑ i = 1 B ( a , b ) i 2 i − a − b Z ( a + b − i , i ) + a ∑ i = 1 A ( a , b ) i 2 i − a − b Z ( i , a + b − i ) − a ∑ i = 1 A ( a , b ) i ∞ ∑ x 2 = 1 ∞ ∑ x 1 = 1 1 + ( − 1 ) x 2 2 1 ( x 1 + x 2 ) i x a + b − i 2 − a ∑ i = 1 A ( a , b ) i 2 − a − b ∞ ∑ x 2 = 1 1 x a + b 2 = b ∑ i = 1 B ( a , b ) i 2 i − a − b Z ( a + b − i , i ) + a ∑ i = 1 A ( a , b ) i 2 i − a − b Z ( i , a + b − i ) − a ∑ i = 1 A ( a , b ) i 1 2 ( Z ( i , a + b − i ) + Z ( i , − ( a + b − i ))) − ( a + b − 1 ) ! ( a − 1 ) ! b ! 2 − a − b Z ( a + b ) , (4.4) 12 with A a , b i = ( a + b − i − 1 ) ! ( a − i ) ! ( b − 1 ) ! (4.5) B a , b i = ( a + b − i − 1 ) ! ( b − i ) ! ( a − 1 ) ! . (4.6) Actually there is a slight problem with the above deriv ation. At two points we changed the summation range. Once from ∞ to ∞ / 2 and once from ∞ to 2 ∞ . This causes no problems if the sum is ﬁnite, but for the diver gent sums thi s needs a correction t erm. The second case is harmless as it concerns only an inner sum, the step in which ( − 1 ) x 2 is introduced. But the ﬁrst case, in t he very ﬁrst st ep of the deriva tion, needs a correction term. Hence the full formula becomes : E ( a , σ b b ) = 1 2 ( Z ( a , σ b b ) + Z ( − a , − σ b b )) = 1 2 δ ( a − 1 ) Z ( − 1 ) Z ( σ b b ) − 1 2 δ ( a − 1 ) δ ( σ b b − 1 ) Z ( − 2 ) + b ∑ i = 1 B ( a , b ) i 2 i − a − b Z ( a + b − i , σ b i ) + a ∑ i = 1 A ( a , b ) i 2 i − a − b Z ( σ b i , a + b − i ) − a ∑ i = 1 A ( a , b ) i 1 2 ( Z ( σ b i , σ b ( a + b − i ) ) + Z ( σ b i , − σ b ( a + b − i ) )) − ( a + b − 1 ) ! ( a − 1 ) ! b ! 2 − a − b Z ( a + b ) . (4.7) Here als o the signs on the indices a and b are included which i s only a very mild com - plication in t he deriv ati on. The function δ ( m ) i s one when m is zero and zero otherwise. The σ -variables ha ve a value that is eith er + 1 or − 1 and indi cate non-alternating and al- ternating sums. Due to the sy mmetry of the starting formula a sign on the ﬁrst var iable is not necessary . If we put it anyway in the form of σ a , σ b will hav e to be replaced by σ a σ b in the right hand side. It is quite rele vant to take these σ factors along. Although they are usually not needed to g et a comp lete coverage of depth d = 2 sums, in the case of greater depth sums they are necessary . The abov e deriva tion shows basically all techni ques we need for the deri v ation of the greater depth formu las. In the sequel we wi ll only carry the σ factors that survive conditions posed during the deriv ation. The deriv ation of the depth 3 form ula follows a simi lar b ut slightl y m ore complicated path. Agai n, we ﬁrst om it the signs of the indices and t he correction terms for div er gent integrals when we double or half the summ ation range. Then we present the compl ete formula. In the deri v ation we wi ll be a bit sh orter this time as the techniques are all similar to what we hav e sho wn above. 13 E ( a , b , c ) = 1 2 ( Z ( a , b , c ) + Z ( − a , − b , c )) = ∞ ∑ x 1 = 1 ∞ ∑ x 2 = 1 ∞ ∑ x 3 = 1 1 ( x 1 + x 2 + x 3 ) a ( x 2 + x 3 ) b x c 3 1 + ( − 1 ) x 1 2 = ∞ ∑ x 1 = 1 ∞ ∑ x 2 = 1 ∞ ∑ x 3 = 1 1 ( 2 x 1 + x 2 + x 3 ) a ( x 2 + x 3 ) b x c 3 = ∞ ∑ x 1 = 1 ∞ ∑ x 2 = 1 ∞ ∑ x 3 = 1 a ∑ i = 1 A ( a , b ) i 1 ( 2 x 1 + 2 x 2 + 2 x 3 ) a + b − i ( 2 x 1 + x 2 + x 3 ) i x c 3 + ∞ ∑ x 1 = 1 ∞ ∑ x 2 = 1 ∞ ∑ x 3 = 1 b ∑ i = 1 B ( a , b ) i 1 ( 2 x 1 + 2 x 2 + 2 x 3 ) a + b − i ( x 2 + x 3 ) i x c 3 = b ∑ i = 1 B ( a , b ) i 2 − a − b + i Z ( a + b − i , i , c ) + a ∑ i = 1 A ( a , b ) i 2 − a − b + i Z ( i , a + b − i , c ) − a ∑ i = 1 A ( a , b ) i 2 − a − b + i K ( 1 ) 1 ( a + b − i , i , c ) − a ∑ i = 1 A ( a , b ) i 2 − a − b + i K ( 1 ) 2 ( i , a + b − i , c ) , (4.8) with the K function s gi ven below . The full formula becomes E ( a , σ b b , σ c c ) = 1 2 ( Z ( a , σ b b , σ c c ) + Z ( − a , − σ b b , σ c c )) = 1 2 Z ( − 1 ) Z ( σ b b , σ c c ) δ ( a − 1 ) − 1 2 Z ( − 2 ) Z ( σ c c ) δ ( a − 1 ) δ ( σ b b − 1 ) + 1 2 Z ( − 3 ) δ ( a − 1 ) δ ( σ b b − 1 ) δ ( σ c c − 1 ) + b ∑ i = 1 B ( a , b ) i 2 − a − b + i Z ( a + b − i , σ b i , σ c c ) + a ∑ i = 1 A ( a , b ) i 2 − a − b + i Z ( σ b i , a + b − i , σ c c ) − a ∑ i = 1 A ( a , b ) i 2 − a − b + i K ( 1 ) 1 ( a + b − i , σ b i , σ c c ) − a ∑ i = 1 A ( a , b ) i 2 − a − b + i K ( 1 ) 2 ( σ b i , a + b − i , σ c c ) . (4.9) 14 The correction terms wit h the δ –functions are d ue to the hal ving of t he summation range in the ﬁrst step. The K –function s are gi ven by K ( 1 ) 1 ( a , σ b b , σ c c ) = ∞ ∑ x 1 = 1 ∞ ∑ x 2 = 1 σ 2 x 1 + x 2 b σ x 2 c ( x 1 + x 2 ) a ( 2 x 1 + x 2 ) b x c 2 = ( − 1 ) b a ∑ i = 1 A ( a , b ) i 2 a − i Z ( i , σ b σ c ( a + b + c − i )) + ( − 1 ) b b ∑ i = 1 B ( a , b ) i 2 a − 1 ( Z ( i , σ b σ c ( a + b + c − i )) + Z ( − i , − σ b σ c ( a + b + c − i ))) + ( − 1 ) b B ( a , b ) 1 2 a − 1 Z ( − 1 ) Z ( σ b σ c ( a + b + c − 1 )) (4.10) K ( 1 ) 2 ( σ a a , b , σ c c ) = ∞ ∑ x 1 = 1 ∞ ∑ x 2 = 1 ∞ ∑ x 3 = 1 σ x 1 + 2 x 2 + x 3 a σ x 3 c ( x 1 + 2 x 2 + x 3 ) a ( x 2 + x 3 ) b x c 3 = ( − 1 ) c 2 b − 1 c ∑ i = 1 B ( b , c ) i ( − 1 ) i ( Z ( σ a a , ( b + c − i ) , σ c i ) + Z ( σ a a , − ( b + c − i ) , − σ c i )) − ( − 1 ) c 2 b − 1 b ∑ i = 1 A ( b , c ) i ( Z ( σ a a , σ c ( b + c − i ) , i ) (4.11) + Z ( σ a a , σ c ( b + c − i ) , − i )) − ( − 1 ) c 2 b − 1 ( b + c − 1 ) ! ( b − 1 ) ! c ! ( Z ( σ a a , ( b + c )) + Z ( σ a a , − ( b + c ))) The last term in the function K ( 1 ) 1 is also a correction term because we ha ve to double the summation range on the Z -function of which the ﬁrst index is one. Because the second index cannot be one in that case, we only need one correc tion term. At depth 4 the relation becomes yet a bit more complicated b ut the deriv atio n follo ws exactly the same path. W e start with appl ying t he non-trivial partial fractioni ng and then we ha ve to try to re write the result s i n terms of Z –functions by percolating the f actors two to t he right . As there is one more sum this t akes another s tep and we get two layers o f 15 K –functions: E ( a , σ b b , σ c c , σ d d ) = 1 2 ( Z ( a , σ b b , σ c c , σ d d ) + Z ( − a , − σ b b , σ c c , σ d d )) = 1 2 Z ( − 1 ) Z ( σ b b , σ c c , σ d d )) δ ( a − 1 ) − 1 2 Z ( − 2 ) Z ( σ c c , σ d d )) δ ( a − 1 ) δ ( σ b b − 1 ) + 1 2 Z ( − 3 ) Z ( σ d d )) δ ( a − 1 ) δ ( σ b b − 1 ) δ ( σ c c − 1 ) − 1 2 Z ( − 4 ) δ ( a − 1 ) δ ( σ b b − 1 ) δ ( σ c c − 1 ) δ ( σ d d − 1 ) + b ∑ i = 1 B ( a , b ) i 2 − a − b + i Z ( a + b − i , σ b i , σ c c , σ d d )) + a ∑ i = 1 A ( a , b ) i 2 − a − b + i Z ( σ b i , a + b − i , σ c c , σ d d )) − a ∑ i = 1 A ( a , b ) i 2 − a − b + i K ( 1 ) 1 ( a + b − i , σ b i , σ c c , σ d d )) − a ∑ i = 1 A ( a , b ) i 2 − a − b + i K ( 1 ) 2 ( σ b i , a + b − i , σ c c , σ d d )) (4.12) 16 K ( 1 ) 1 ( a , σ b b , σ c c , σ d d ) = ∞ ∑ x 1 = 1 ∞ ∑ x 2 = 1 ∞ ∑ x 3 = 1 σ 2 x 1 + x 2 + x 3 b σ x 2 + x 3 c σ x 3 d ( x 1 + x 2 + x 3 ) a ( 2 x 1 + x 2 + x 3 ) b ( x 2 + x 3 ) c x d 3 = ( − 1 ) b a ∑ i = 1 A ( a , b ) i 2 a − i Z ( i , σ b σ c ( a + b + c − i ) , σ d d ) + ( − 1 ) b b ∑ i = 1 B ( a , b ) i 2 a − 1 ( Z ( i , σ b σ c ( a + b + c − i ) , σ d d ) + Z ( − i , − σ b σ c ( a + b + c − i ) , σ d d )) + ( − 1 ) b B ( a , b ) 1 2 a − 1 Z ( − 1 ) Z ( σ b σ c ( a + b + c − 1 ) , σ d d ) (4.13) K ( 1 ) 2 ( σ a a , b , σ c c , σ d d ) = ∞ ∑ x 1 = 1 · · · ∞ ∑ x 4 = 1 σ x 1 + 2 x 2 + x 3 + x 4 a σ x 3 + x 4 c σ x 4 d ( x 1 + 2 x 2 + x 3 + x 4 ) a ( x 2 + x 3 + x 4 ) b ( x 3 + x 4 ) c x d 4 = ( − 1 ) c 2 b − 1 c ∑ i = 1 B ( b , c ) i ( − 1 ) i ( Z ( σ a a , ( b + c − i ) , σ c i , σ d d ) + Z ( σ a a , − ( b + c − i ) , − σ c i , σ d d )) + ( − 1 ) c b ∑ i = 1 A ( b , c ) i 2 b − i Z ( σ a a , σ c ( b + c − i ) , i , σ d d ) − ( − 1 ) c b ∑ i = 1 A ( b , c ) i 2 b − i K ( 2 ) 1 ( σ a a , σ c ( b + c − i ) , i , σ d d )) − ( − 1 ) c b ∑ i = 1 A ( b , c ) i 2 b − i K ( 2 ) 2 ( σ a a , σ c ( b + c − i ) , i , σ d d )) (4.14) K ( 2 ) 1 ( σ a a , σ b b , c , σ d d ) = ∞ ∑ x 1 = 1 ∞ ∑ x 2 = 1 ∞ ∑ x 3 = 1 σ x 1 + 2 x 2 + x 3 a σ 2 x 2 + x 3 b σ x 3 d ( x 1 + 2 x 2 + x 3 ) a ( 2 x 2 + x 3 ) b ( x 2 + x 3 ) c x d 3 = ( − 1 ) d c ∑ i = 1 A ( c , d ) i 2 c − i Z ( σ a a , σ b σ d ( b + c + d − i ) , i ) + ( − 1 ) d 2 c − 1 d ∑ i = 1 B ( c , d ) i ( − 1 ) i ( Z ( σ a a , ( b + c + d − i ) , σ b σ d i ) + Z ( σ a a , − ( b + c + d − i ) , − σ b σ d i )) − ( − 1 ) d 2 c − 1 c ∑ i = 1 A ( c , d ) i ( Z ( σ a a , σ b σ d ( b + c + d − i ) , i ) + Z ( σ a a , σ b σ d ( b + c + d − i ) , − i )) − ( − 1 ) d 2 c − 1 ( c + d − 1 ) ! ( c − 1 ) ! d ! ( Z ( σ a a , ( b + c + d )) + Z ( σ a a , − ( b + c + d )) ) (4.15) K ( 2 ) 2 ( σ a a , σ b b , c , σ d d ) = ∞ ∑ x 1 = 1 · · · ∞ ∑ x 4 = 1 σ x 1 + x 2 + 2 x 3 + x 4 a σ x 2 + 2 x 3 + x 4 b σ x 4 d ( x 1 + x 2 + 2 x 3 + x 4 ) a ( x 2 + 2 x 3 + x 4 ) b ( x 3 + x 4 ) c x d 4 17 = ( − 1 ) d 2 c − 1 d ∑ i = 1 B ( c , d ) i ( − 1 ) i ( Z ( σ a a , σ b b , ( c + d − i ) , σ d i ) + Z ( σ a a , σ b b , − ( c + d − i ) , − σ d i )) + ( − 1 ) d c ∑ i = 1 A ( c , d ) i 2 c − i Z ( σ a a , σ b b , σ d ( c + d − i ) , i ) − ( − 1 ) d 2 c − 1 c ∑ i = 1 A ( c , d ) i ( Z ( σ a a , σ b b , σ d ( c + d − i ) , i ) + Z ( σ a a , σ b b , σ d ( c + d − i ) , − i ) − ( − 1 ) d 2 c − 1 ( c + d − 1 ) ! ( c − 1 ) ! d ! ( Z ( σ a a , σ b b , ( b + c )) + Z ( σ a a , σ b b , − ( b + c ))) . (4.16) When we do depth 5 we see that, like K ( 1 ) 2 , also the K ( 1 ) 1 splits off t wo ne w functions. Hence to produce a generic routine for any d epth we hav e to look at a few ver y general steps. In the general case the equati ons (4.12, 4.13) and (4.15) stay more or less the same. They jus t get more indices to t he right. The difference comes with the equat ions for K ( 2 ) . W e have to make a distinction whether there are still many indices to the right or whether we are termi nating. Th e terminatin g equati ons are also m ore or less the sam e as the equations for K ( 2 ) above, but no w with more indices to the left. Th is lea ves the ‘intermediary’ objects: K ( i ) 1 ( M , σ a a , b , σ c c , N ) = ∞ ∑ x 1 = 1 ∞ ∑ x 2 = 1 σ x M + 2 x 1 + x 2 + x N a σ x 2 + x N c ( x M + 2 x 1 + x 2 + x N ) a ( x 1 + x 2 + x N ) b ( x 2 + x N ) c (4.17) K ( i ) 2 ( M , σ a a , σ b b , c , σ d d , N ) = ∞ ∑ x 1 = 1 ∞ ∑ x 2 = 1 σ x M + 2 x 1 + x 2 + x N a σ 2 x 1 + x 2 + x N b σ x 2 + x N d ( x M + 2 x 1 + x 2 + x N ) a ( 2 x 1 + x 2 + x N ) b ( x 1 + x 2 + x N ) c ( x 2 + x N ) d . (4.18) In th ese formu las M and N i ndicate a range of ind ices. There are m ore su ms and fac tors in the numerator and denominator , but w e just om it them as t hey do not take part in the ‘action’. W e use t he same techniques applied before to move the factor 2 that m ultiplies x 1 to x 2 , to th e right. When N i s empty we run in to a termination conditi on and switch to the equations for K ( 2 ) , K ( i ) 1 (( M ) , σ a a , b , σ c c , ( n 1 , N )) = ( − 1 ) c b ∑ i = 1 A ( b , c ) i 2 b − i Z ( M , σ a a , σ c ( b + c − i ) , i , n 1 , N ) +( − 1 ) c c ∑ i = 1 B ( b , c ) i 2 b − 1 ( − 1 ) i ( Z ( M , σ a a , ( b + c − i ) , σ c i , n 1 , N ) + Z ( M , σ a a , − ( b + c − i ) , − σ c i , n 1 , N )) 18 − ( − 1 ) c b ∑ i = 1 A ( b , c ) i 2 b − i K ( i ) 1 (( M , σ a a ) , σ c ( b + c − i ) , i , n 1 , ( N )) − ( − 1 ) c b ∑ i = 1 A ( b , c ) i 2 b − i K ( i ) 2 (( M ) , σ a a , σ c ( b + c − i ) , i , n 1 , ( N )) (4.19) K ( i ) 2 (( M ) , σ a a , σ b b , c , σ d d , ( n 1 , N )) = ( − 1 ) d c ∑ i = 1 A ( c , d ) i 2 c − i Z ( M , σ a a , σ b σ d ( c + d − i ) , i , n 1 , N ) +( − 1 ) d d ∑ i = 1 B ( c , d ) i 2 c − 1 ( − 1 ) i ( Z ( M , σ a a , σ b ( b + c + d − i ) , σ d i , n 1 , N ) + Z ( M , σ a a , − σ b ( b + c + d − i ) , − σ d i , n 1 , N )) − ( − 1 ) d c ∑ i = 1 A ( c , d ) i 2 c − i K ( i ) 1 (( M , σ a a ) , σ b σ d ( b + c + d − i ) , i , n 1 , ( N )) − ( − 1 ) d c ∑ i = 1 A ( c , d ) i 2 c − i K ( i ) 2 (( M , σ a a , ( b + c + d − i ) , σ b σ d i , n 1 , ( N )) . (4.20) As one can see, each step of the iteration dim inishes N b y one unit ( n 1 is an index with it s sign) and M may or may not get one more index. The above formulas can be programmed rather easily and compactly in a language like FORM . W e hav e ﬁrst programmed and tested the cases 2, 3 , 4, 5 and after that we hav e made a generic routine that can handle any depth. Also this routi ne has been tested exhaustiv ely . It can be found in the library . 4.2 The Role of the Generalized Doubling Relations Let us start wi th a mo diﬁcation of th e program for expressing Euler sums i nto a m inimal set t hat w as used for testing TFORM [22]. It was modiﬁed, so as to allow running only with sums/functio ns up to a give n depth. W e use the same relation s, up to that d epth, as in the complete program, i.e. we use t he stuf ﬂes, t he shuf ﬂes and t he doubling r elations, b ut not the GDRs. This should generate ne w information because one is often interested in su ms of limited depth but lar ge weight. When we compare t he number of remaini ng variables with the conjectures [12, 13], we note that in many cases we ha ve more v ariabl es left. Howe ver , if we increase the depth these remaining var iables are eliminated after all. W e set up the program in such a way that these objects may be recognized easily . In T able 1 we present how many of these constants are left and at which depth. T able 1 ind icates that there must be a signiﬁcant ‘leaking’ of relations at greater depths that create nontrivial resul ts at lower depth. As an example we deriv ed t he d = 2 relati on at weight 6 w ithout s ubstituti ng the lo wer weigh t constants and kee ping track of all p roducts of lower weight objects that combined in shuf ﬂes and in stufﬂes. The relation we refer t o is giv en as Eq. (27) in Ref. [12] : Z − 4 , − 2 ( ∞ ) = − H − 4 , 2 ( 1 ) = 97 420 ζ 3 2 − 3 4 ζ 2 3 . (4.21) 19 weight depth number type 6 2 1 d = 2 6 3 1 d = 2 6 4 0 6 5 0 6 6 0 7 3 1 d = 3 7 5 0 8 2 1 d = 2 8 4 2 d = 2, d = 4 8 6 0 9 3 3 3 × ( d = 3 ) 9 5 2 d = 3, d = 5 9 7 0 10 2 2 2 × ( d = 2 ) 10 3 2 2 × ( d = 2 ) 10 4 6 2 × ( d = 2 ) ,4 × ( d = 4 ) 10 5 6 2 × ( d = 2 ) ,4 × ( d = 4 ) 10 6 3 d = 2, d = 4, d = 6 10 8 0 T able 1: Num ber of constants remaining when runni ng at ﬁxed depth for a g iv en weight. W ith ﬁxed depth we mean all depths up to the gi ven value. depth shufﬂ es stuf ﬂes 2 11 8 3 52 19 4 72 41 T able 2: Number of shufﬂes and stufﬂes separated by depth cont ributing to equation (4.21). The results are shown i n T able 2. W e see that a t otal of 203 equati ons make contributions to the ﬁnal result. Considering this, it sho uld not come as a great surprise th at attempts to deriv e t his equation by hand using shufﬂe a nd stufﬂ e relations ha ve fa iled thus far . It is of course possible to obtain this result by dif ferent means as was shown in re f [26] where the ﬁnit e harmonic sum S − 4 , − 2 ( N ) was calculated in terms of the following one- dimensional integral representation: S − 4 , − 2 ( N ) = − M  4Li 5 ( − x ) − ln ( x ) Li 4 ( − x ) x − 1  +  ( N ) (4.22) + 1 2 ζ 2 [ S 4 ( N ) − S − 4 ( N )] − 3 2 ζ 3 S 3 ( N ) + 21 8 ζ 4 S 2 ( N ) − 15 4 ζ 5 S 1 ( N ) , 20 where M [ f ( x )]( N ) = Z 1 0 d x x N f ( x ) . (4.23) Since Z 1 0 d x 4 [ Li 5 ( − x ) + ( 15 / 16 ) ζ 5 ] − ln ( x ) Li 4 ( − x ) x − 1 = − 811 840 ζ 3 2 + 3 4 ζ 2 3 (4.24) one obtains with Z − 4 , − 2 = lim N → ∞ S − 4 , − 2 ( N ) − ζ 6 (4.25) the above result . It should, ho we ver , be clear that if s uch m ethods are needed to replace the phenomenon of leakage, it will b e a near impossibilit y to go to much greater v alues of the weight parameter . Using the GDRs at depth d = 2 resolves the problem completely . Only the depth d = 2 shufﬂe s and stufﬂ es in combination with these GDRs giv e al ready the desired formula. T o study t he p roblem at d epth d = 3 , we recreated an old program b y one of us 6 that only de termines r elations at leading d epth for objects of whi ch the index ﬁeld is a L yndon word. The FORM version of the program is rather fast when applied at depth d = 3 , see T able 3. W e see a steady increase i n the numb er of un determined constants. In T ables 3, 4 we list under ‘expected’ the n umber of undetermined constants according to conjecture [12]. The results for the weights 7 and 9 are in agreement with the numbers in T able 1. T o see whether we could im prove the situation, we tried programming generalizations of the formulas D 0 and D 1 of Ref. [48]. They m ade n o difference. Close inspection re veals that the formula D 0 is ano ther form of the shuf ﬂe formulas with the c ombinatorics included properly . T he formul a D 1 , or Markett form ula [49], also d oes not add anything new . It seems to be a combination of shufﬂes and stufﬂes. Next we applied the GDRs at depth d = 3 and these reduce the n umber o f undetermined constants to their expected value. This m eans t hat if we in clude the GDRs we can run the program at maximum depth d = 3 and get a complete set of expressions for all depth d = 1 , 2 and 3 o bjects. At the m oment we ha ve ver iﬁed this for all weights up to w = 51 . The run for the hi ghest weight took about 20 hours of CPU time on a single Xeon processor at 2.33 GHz. W e ha ve made a simil ar program for depth d = 4 . This is of course m uch slower and hence we cannot go to such large v alues for the weight. Th e results are given in T able 4. Ag ain we see an increase in the number of extra undetermined objects and again application of the GDRs resolved the issue. The phenom enon of leakage is rather messy . Basically equations that are in nat ure of a greater depth have to comb ine ﬁrst to elim inate mos t objects of th is depth . After th is a fe w equations remain between l ower d epth objects. Such leakage is im possible without the stufﬂe relations. The sh uf ﬂe relations by themselves do not give t erms with a lower 6 The program had an error and hence ga ve ris e to a wrong conjectu re. 21 weight constants expected 5 1 1 7 3 2 9 6 3 11 11 5 13 17 7 15 23 9 17 32 12 19 41 15 21 51 18 23 63 22 25 76 26 27 89 30 29 105 35 31 121 40 33 138 45 35 157 51 37 177 57 39 197 63 41 220 70 43 243 77 45 267 84 47 293 92 49 320 100 51 347 108 T able 3: Remaining con stants at depth d = 3 compared to the number of expected con- stants. weight constants expected 6 1 1 8 3 2 10 9 5 12 21 8 14 39 14 16 66 20 18 102 30 20 149 40 22 209 55 T able 4: Remaining con stants at depth d = 4 compared to the number of expected con- stants. 22 weight no doubling no GDRs 8 1 0 10 1 0 11 2 1 12 3 1 T able 5: Num ber of excess elements wh en no doubl ing relations (also no GDR s) are used, and when only no GDRs are used. depth and neit her do the relatio ns based on the doubling formula. But whether these extra relations come from t he stufﬂes alone or materialize only after com bining stufﬂ es and shufﬂe s, and m aybe d oublings, is currently not clear . What is clear is that they in volv e a very large number of equations. In all cases whi ch we studied the leakage goes over at least two units of depth. This makes it very di f ﬁcult to in vestigate. Fortunately the GDRs seem to resolve these problems. W e formulat e Conjectur e 1 : The st ufﬂ e, shufﬂe, dou bling and Generalized Doubling Relations are sufﬁ cient to reduce the Euler sums of a given weigh t and depth to a minimal set that is in agreement with the conjecture [12], both in weight and in depth. ✷ Even if we could disp ense with the GDRs up to weight w = 10 , the wh ole situatio n changes at weight w = 11 , see T abl e 5. Running only stufﬂes, shufﬂ es and doubling relations l ea ves one variable in exce ss of the conjecture [12]. The GDRs provide th e missing equation by which this variable is expressed in terms of the other remaining var iables and agreement with conj ecture [12] is reached. The same effect occurs at weight w = 12 . A gain there is one variable too many if the GDRs a re not used. W e cannot check this beyond weight w = 12 , because leakage f orces us t o run all depths for a given weight if we e xclu de the GDRs. This becomes e xcessive in terms of curr ent computer re sources. Alternative ly on e could h a ve used th e relati ons of equation (2.26) to resolve thi s is sue, but these relations do not h elp with the p roblem of runni ng at a li mited depth. Hence we hav e to add the GDRs anyway . 5 The Compute r Program W e ha ve combined the above relations in to a ne w computer program to resolve all re- lations between MZVs and reduce them to a minimal set . In principle this is done by writing down all equatio ns for the MZVs of a giv en weight and t hen solvi ng the system. A few variables at the given weight may remain and there will be products of objects of lower weigh t. Considering the size of the probl em and its sparsity it did not look to u s like a typical problem to solve by matrix techniques e ven though other people ha ve do ne so [50, 51]. T ypically there would be many thousands of zeroes for each non-zero element. The ad- vantage of com puter alg ebra is t hat i n a sp arse p olynomial representation tho se zeroes will not be present and need no attention. Hence we have selected a rather special m ethod the 23 essence of which has already been used in references [10, 18, 22], alt hough not described there in d etail. W e select the FORM system, beca use it is by far the best suited for t his kind of probl ems. Since we go to m uch greater weights th an p re viously in vestigated, we take the opportunit y to give here a better description of the completely renewed version of the program. W e start generating a master expression which con tains one term for each sum that we want t o compute. For the MZVs of weight w = 4 this e xpression looks in computer terms like FF = +E(0,0,0,1 )*(H(0,0,0 ,1)) +E(0,0,1,1 )*(H(0,0,1 ,1)) +E(0,1,0,1 )*(H(0,1,0 ,1)); W e have used already that we will only compute the ﬁnit e elements and that there is a duality that allows us t o elim inate all elements with a d epth greater than h alf t he weight. When the depth is e xactly half the weight we choose from a sum and its dual the element that comes ﬁrst lexicographically . W e work in terms of t he H -functions because for the Euler sums the b asis of reference [12] turns out to b e ideal. T his basis consists of all L y ndon words of negativ e odd integers that add up in absolute v alue to the weig ht. For the MZVs these H -f unctions and the Z -functions are identical anyway and hence we could keep a single program for most procedures. W e pull the function E outside b rackets. The contents of a brack et is what we kno w about th e object indicated by the indices of the functi on E. In the beginning this i s all trivial kno wledge. Assume now t hat we generate the stufﬂe relati on H 0 , 1 H 0 , 1 = H 0 , 0 , 0 , 1 + 2 H 0 , 1 , 0 , 1 (5.1) The left hand side can be subst ituted from the t ables for the lo wer weight M ZVs. Hence it becomes ζ 2 2 . In the program ζ 2 is called z2 . The right hand side objects are replaced by the contents of the correspondin g E brackets in the master expression. These are for now trivial substitutions. From the result we generate the substi tution id H(0,1,0,1) = z2ˆ2/2-H(0,0,0, 1)/2; which we apply to the master expression. Hence the master expression becomes FF = +E(0,0,0,1 )*(H(0,0,0 ,1)) +E(0,0,1,1 )*(H(0,0,1 ,1)) +E(0,1,0,1 )*(z2ˆ2/2- H(0,0,0,1) /2); Let us now genera te the corresponding shufﬂ e relation: H 0 , 1 H 0 , 1 = 4 H 0 , 0 , 1 , 1 + 2 H 0 , 1 , 0 , 1 (5.2) 24 and replace the right hand side objects by the c ontents of the corresponding E bra ckets in the master expression. This giv es ζ 2 2 = 4 H 0 , 0 , 1 , 1 + ζ 2 2 − H 0 , 0 , 0 , 1 (5.3) which leads to the substitut ion id H(0,0,1,1) = H(0,0,0,1)/4; and we obtain FF = +E(0,0,0,1 )*(H(0,0,0 ,1)) +E(0,0,1,1 )*(H(0,0,0 ,1)/4) +E(0,1,0,1 )*(z2ˆ2/2- H(0,0,0,1) /2); W e also need the diver gent shu f ﬂes and stuf ﬂes. This is done by including the shufﬂ es in volving the basic diver gent object and breaking d own the multiple div er g ent sums with the stufﬂe relati ons as in: H 1 H 0 , 0 , 1 = 2 H 0 , 0 , 1 , 1 + H 0 , 1 , 0 , 1 + H 1 , 0 , 0 , 1 = − H 0 , 0 , 0 , 1 + H 0 , 0 , 1 , 1 + H 0 , 1 , 0 , 1 + H 1 H 0 , 0 , 1 ; (5.4) In the case we use H 1 as the only diver gent obj ect, this is equiv alent to usi ng Hoff- mann’ s [52] relation. W e can use any combi nation i n volving diver gent objects, provided not both are di ver gent simultaneously . Substit uting from the master e x pression we get the relation 0 = − 5 4 H 0 , 0 , 0 , 1 + 1 2 ζ 2 2 (5.5) and hence the substit ution id H(0,0,0,1) = z2ˆ2*2/5; and ﬁnally the master expression becomes FF = +E(0,0,0,1 )*(z2ˆ2*2/ 5) +E(0,0,1,1 )*(z2ˆ2/10 ) +E(0,1,0,1 )*(z2ˆ2*3/ 10); Now we can read off the values of all MZVs of weight 4 that we set out to comp ute. All other elem ents can be obtained from these by tri vial opera tions th at inv olve the use of one or two relations only . The metho d should be clear now: we generate the mast er expression that contains all nontrivial objects that we need to comput e. Then we generate all known equations one by one, pu tting in the kno wledge th at is con tained in the master e xpression. After that we incorpo rate th e new kn owledge in th e m aster expression (provided t he equat ion does 25 w/g 64 128 256 512 1024 2048 4096 9 62 56 61 10 477 406 442 11 5826 4651 3799 3623 5157 12 65591 50926 62867 T able 6: Execution times i n seconds for Eu ler s ums at any dept h as a function of weigh t and the size of the groups in the Gaussian elimi nation scheme. All runs were with TFORM on an 8 Xeon-cores machine at 3 GHz. not become trivial which will happ en frequently , because we have more equati ons t han var iables). W ith this m ethod we d o no t need all equati ons to be in memory s imultaneousl y . But there is a very important observation: the order in which the equati ons are generated will determine the size the m aster expression can hav e during t he calculation. This intermedi- ate expression swell should be controlled as much as possible, bec ause it can make many orders of magnitude diff erence in t he ex ecution time and t he space n eeded. And there is another problem: substitut ing a n e w equation in the master expression can be rather costly when thi s expression becomes rather big. T o have to d o this each tim e is wa steful because the m aster expression will hav e to be brought to normal order again. Therefore we have a dopted a scheme i n which we generate the equation s in groups. Then we apply ﬁrst a Gauss ian elimination scheme among t he equations in the group , elim inating both above and below the diagonal. If we ha ve G equations l eft we can subst itute G variables in the master expression s imultaneousl y . Again, this is not optimal yet as that w ould giv e G substitution stat ements and hence each t erm needs G pattern m atchings. T o improve upon this we ent er these G obj ects in a temporary table and t he substitut ion in the master expression is by a single table lookup. T his is a bin ary search inside FORM and hence wh en we hav e g rouped for instance 512 equations, the lookup takes only 9 compares, each of which is anyway much faster than a full patt ern matching . The difference shows in a run we made on a machine wit h a single Op teron processor . When running t he equations for MZVs one by one at weight 18 , th e run took 26761 sec, whi le with groups of 256 equa- tions the sam e program ran in 2974 sec. Over th e range in weights that we experimented with, the op timal group size we found for the MZVs was close to 2 ( w − 1 ) / 2 . This is the value we us e in the program. For th e Euler sums t he best va lue ob eys a m ore in volv ed relation because the num ber of v ariables goes wi th a po wer of three. W e ha ve m easured the effect and it is shown in T able 6. From this T able it lo oks like a decent value for the size of the g roups is 2 3 w / 2 − 7 in which the exponent is rounded down to the nearest in teger . W e see, howe ver , that the exact v alue is not very critical. If it would be of great im portance to improve over thi s scheme, on e could set up a tree structure in the Gauss ian scheme. This would change it s quadratic (in the size of t he groups) nature to a G log ( G ) beha viour . It would, howe ver , make the code much more complicated and anyway , this is n ot wh ere currently most computer time is used. As a consequence we decided to stay with the simple grouping. This leav es determining a good order in which to generate the equations. It requires much tri al and error and we are not cl aiming that we have the best schem e possib le. The 26 scheme for the s tufﬂ es is rather good, b ut for the sh uf ﬂes it could probably be better . Once we c ould run what we w anted to run, we ha ve s topped searching intensive ly . Anyway , t he intermediate expression swell is rather m oderate as is sh own i n the T ables containin g the results below . Before we discuss the order of the equations, we make se veral o bservations: • Shuf ﬂes preserve depth. • Stufﬂ es either preserve depth or lower it . • The number of indices that are one in sum notation is either preserve d or lo wered by stufﬂes. • The shufﬂe relation s can contain many more ter ms than the stufﬂe relations. • The shu f ﬂes (which are exe cuted in integral notation) can contain large combina- toric factors when there are long sequences of zeroes or ones. This lowers the number of terms in the equation. Based on the abov e observ ati ons we s tart with the equ ations with the lo west dept h, and then do the ones wit h the next depth, etc. In the case th at we only look at t he MZVs, we only need t o go up to half the weight (rounded down), because the duality relation takes care of the other sums. In the case of the Euler sums we hav e t o go ‘all the way’. For each depth we do ﬁrst the stufﬂes and then the s hufﬂ es. There are actually con- jectures about t hat one does not need all s tufﬂ es but only a limit ed subset. W e do not use these conj ectures because they would make it necessary to apply more shuf ﬂe relations and tho se are more comp licated than the s tufﬂ es that we would om it. W e hav e veriﬁed experimentally that this w o uld make the program signiﬁcantly slower . In the case of Eul er s ums we have two more categories of equatio ns: t he equations due to the doub ling relation and the equations due to t he GDRs. It looks l ike we do not need all equations from the latter category , but because they are not extremely costl y , we hav e not been mo tiv ated enough to run m any programs testin g what c an be done here. W e just run them all and this way there is no r isk that we omit something essential. They a re, howe ver , more c ostly than the shuf ﬂe equations and hence we put them after the sh ufﬂ es. But more ordering wit hin the group of (generalized) doubling equ ations is not rele vant as there are only comparative ly fe w substitu tions genera ted by them. T o deal with the stufﬂes at a giv en weight and depth we generate an expression that contains one term for each s tufﬂ e relati on t hat we will use. Then we appl y seve ral o per - ations that multipl y each term with a function with arguments based on th e equation to be generated. T he effect of this is that at the next sorting the equations will be ordered according to t hese arguments. This can be d one in a rather ﬂexible way . The ordering is in sum notation according to: • The number of indices that are one. • Next comes the number of indices that are tw o, then three etc. • The number of indices in the sum with the smallest depth. 27 • The largest ﬁrst index in either of the two sums. This relatively simple ordering is amazingly effecti ve. When we compute the size of the basis, using arithmetic ove r a 3 1-bit prime number , it giv es a n early monotonically increasing size in th e master expression, indicating that it wi ll be ve ry hard to improve upon i t. Once we have this expression we use a feature of FORM that allows one t o deﬁne a loop in which the l oop var iable takes a v alue which is (sequentially) each time a term from a given e xpression. This way we can now create expressions for each equatio n and each tim e we have enou gh equations to ﬁ ll a group we call the routine that will e xp and the equations and pro cess them. W e do n ot consider stu f ﬂe equations that contain a diver gent sum. Thos e are tak en into consi deration an yway when we ha ve to extract the diver gences in the shufﬂe equations, and for the Euler sums the GDRs. For t he shufﬂes things are mo re com plicated. Again we generate an expression for all shufﬂes for the giv en depth. In this case we generate howe ver only those objects t hat correspond to shu f ﬂes in which one of the objects is only of dept h one. This seems to be sufﬁ cient. W e ha ve ne ver run across a case where the other shuf ﬂes had any additional ef- fect. It is actually possible to restrict the number of shuf ﬂe equations e ven more, although this is only based on conjectures and experimentati on. A formal proof is mi ssing. The ordering is now done acc ording to • The weight of the object of depth one. • The number of indices that are one in sum notation. • For each sum we com pute t he sum of the squares of th e i ndices i n sum not ation. W e order by the maxi mum of either of the two. The biggest comes ﬁrst. • W e select which of t he two su ms has t he small est ﬁrst index. The lar g er values for this number come ﬁrst. • W e add the ﬁrst ind ices of the two sums. T he lar ger va lues come ﬁrst. The complicating factor here is that we ha ve to keep di ver gent sums. W e only keep those equations in which at m ost one obj ect is diver gent, and there is only a single div er gence. Hence sums that hav e the ﬁrst two indices equal to one are not considered. According to o bservation the shufﬂe equations that fulﬁll all following requirements alwa ys reduce to trivial (0 = 0) equations: • The combined depth is at least three. • There is at least one index that is equal to one. • The depth one object has at least weight two. • If the depth one object has weight w = 2 , there are at least two indices equal to one in the other object. 28 MZVs Rational arithmetic Complete results W = 21 MZVs Modular arithmetic Basis only W = 26, D = 8 Figure 1 : Performance of the program. On the x -axis we ha ve th e number of th e m odule in which one group of equations is substituted and on the y -axis the size of the expre ssion at the end of the module (arbitrary units). Th e spikes are due to the shufﬂ es. Harmonic su ms wit h all the same in dex decompose algebraically into a pol ynomial of single harmonic s ums. It is easily shown that the algebraic relation s [37] always allo w to write any h armonic s um in terms of p olynomials of S 1 ( N ) and sum s, which con verge in the limit N → ∞ . All t he above greatly reduces the nu mber of shufﬂe equations that ha ve to be e v aluated. Because this e va luation is one of the expensi ve steps, it speeds up the program signiﬁcantly . On the other hand , it is only an observation made in runs that do not inv olve the greatest weights. For the mo re criti cal runs 7 we h a ve left these equatio ns activ e and spent the e xtra computer time. The above describes the basic program . At thi s point we sp lit it in several varieties. T o ﬁrst determine whether shufﬂes and stuf ﬂes are suf ﬁcient to reduce all MZVs to a basis of the conjectured size, we hav e m ade the simpliﬁcations: • All products of lower weigh t objects are set to zero. This means we w ill only determine whether reduction to a L yndon basis take s place. • W e work modulus a 31-bit prime. W e ha ve als o made runs ove r the rational n umbers. This becomes only problematic for the very highest v alues of the weight. For con structing tables of all sums at a given weight we run the full program. The performance of the program is shown in Figure 1 for a complete run at weight w = 21 and a run to depth d = 8 at weig ht w = 26 . W e see that t he stu f ﬂes give a steady growth 7 W ith this we mean the programs that determ ine the size of the basis when using ar ithmetic over a prime number . Once we ha ve established this, any furth er run s to for instance determin e all values over the rational numbers, we can safely drop these equation s. 29 of the master expression b ut t hat t he shuf ﬂes cause intermediate expression swell which is worse when the depth is much less than half the weig ht. T he result is that when we run the complete system most time is spent with the stuf ﬂe relations while for the limited depth runs by far most time is spent with the shuf ﬂe re lations. In the case of Euler sums the master expression is created with a three letter al phabet ( − 1 , 0 , 1 ) rather than the t wo letter alphabet (0,1) for th e MZVs. In addition there are many more equations to consider because the num ber of lo wer weight objects that we c an multiply eith er b y shu f ﬂes or stufﬂes is correspondi ngly greater . Of course also for the Euler sums it is possible to just study the basis. In addit ion it is possible t o study sum s to a limited depth. This way we ca n go to m uch greater v alues of the weight. This is of course only possible if we use a basis in which the concept of depth is rele vant, like the basis of the odd ne gativ e indi ces that form a L yndon word. W it hout such a basis the calculations become much harder . When we are const ructing tabl es we cannot go qui te as far in w eight as wh en we are determining rank deﬁcienc y . When we use a L yn don basis, t he majority o f terms consists of products of basis elements of lower weigh ts. T his means that we hav e m any mo re terms to carry around. W e observe, in addition, that the coef ﬁcients cont aining the most digits are in the terms with po wers of ζ 2 . Thi s is to be e xpected since ζ N 2 is our reposit ory for all terms of the form ζ m 2 a ζ n 2 b with N = ma + nb . The representation we have selected, together with the modular arithmetic, makes for a very fast treatment of the terms. Th is is reﬂected in the number of terms that can be processed. In one run, which took more than 30 days t he program generated a tot al of more than 7 · 10 12 terms. This seems to be a new re cord. 6 The Running of the Programs W e hav e used the programs of the previous Section to obtain results t o as high a weight and depth as possible, b oth for MZ Vs and Euler sums . Before we start discussing these results we show the parameters of these runs to giv e the reader an impression of what is a va ilable and why there are limitatio ns t o obtain more. W e st art with t he Euler sums. W e h a ve ﬁrst run the comp lete system for the giv en weights, see T able 7. This means that for w = 12 there are expressions for all 236196 Euler sums with that weight, all expressed in terms of the basis of L yndon words of t he negati ve odd integers, see Appendi x A, which is the basi s we use for all Euler sums, unless mentioned differently . The columns m arked ‘variables’ m entions how many variables there are at the start of the program. ‘Remaining’ tells how many basis elements remain in t he end. Under ‘output’ we give the size of the output e xp ression in te xt form at. The column ‘ size’ refers to t he lar gest size of the master expression during the calculat ion. T i me refers to real time to run the program. If the colu mn ‘CP U time’ is present it refer s to the total C PU time by all processors. W e not ice that comput er time is not the issue here, s ee T able 7 8 . The size 8 The ﬁrst time we ran th e w = 12 case on an 8-co re Xeon m achine at 2.33 GHz the run took two full weeks. It just shows ho w good a test case this problem is. Both (T)FOR M and the MZV pro gram ha ve been improved greatly dur ing this project. 30 w var iables eqns remaining size output time [sec] 4 36 57 1 4.3K 2.0K 0.06 5 108 192 2 21K 8.9K 0.12 6 324 665 2 98K 42K 0.37 7 972 2205 4 472K 219K 1.71 8 2916 7313 5 2.25M 1.15M 7.78 9 8748 23909 8 11M 6.3M 50 10 2624 4 77853 11 5 8M 36M 353 11 7873 2 251565 18 360 M 213M 3266 12 236196 809177 25 3.1G 1.29G 47 311 T able 7: Runs on an 8 -core Xeon comput er at 3 GHz and with 32 Gby tes of memory . The column ‘eqns’ gives the number of equations that was considered. weight constants running time [sec] output [Mbyte] 9 956 7 0.26 10 1412 13 0.64 11 1996 24 1.25 12 2724 39 3.18 13 3612 68 5.04 14 4676 10 8 17.1 15 5932 19 9 17.1 16 7396 43 6 71.1 17 9084 60 2 54.9 18 11012 1323 275.9 19 13196 2761 157.1 20 15652 5424 877 21 18396 14090 395 22 21444 21875 2559 T able 8: Summ ary of th e runs at d = 4 . The runs were performed on a computer with 8 Xeons at 3 GHz, using TFORM . of th e results becomes t he m ajor problem. Th is is one of the reasons why we stopped at w = 12 . T echni cally the run at w = 13 is fe asible as it should tak e o f the order of 10 days. The output is, howe ver , projected at almost 8 Gbytes which we considered exc essiv e. W e ha ve also run programs that go to a maximum value of th e depth. T his in volves only a subset of the Euler sum s of that weight and hence s uch programs are much faster . As a consequence we can go to much greater values of the weig ht. In T able 8 we show the s tatistics of the runs up t o depth d = 4 . These are full runs i n the sens e that they are over the r ational numbers and we have kept all terms, inclu ding the products of lower weig ht objects. The dependence on the parit y of the we ight for the higher values is due to the fac t that we run up t o an even depth and the i ndependent variables we use ha ve an even depth for e ven weights and an o dd depth for o dd weights. This means for instance that the depth 31 weight constants remaining running time [sec] output [Mbyte] 9 3394 7 27 1.15 10 5702 7 72 3.11 11 9042 13 172 8.5 12 13686 11 478 20.9 13 19938 22 1330 68.9 14 28134 17 4306 133 15 38642 35 27607 473 16 51862 24 110336 6 88 17 68226 55 450462 2 767 T able 9: Summary of the runs at d = 5 . Same computer as used in T able 8. weight constants running time [sec] output [Mbyte] 14 4676 35 1.3 16 7396 10 5 2.9 18 11012 323 6.0 20 15652 939 11.3 22 21444 2211 20.5 24 28516 5335 35 26 36996 13127 57 28 47012 47056 89 30 58692 100813 137 T able 10: Summary of the runs at d = 4 in mo dular arith metic, dropping all terms that are products of lower weig ht objects. 4 objects for w eight w = 17 can all be expressed in terms of depth d = 3 ob jects. The results for the depth 5 runs are summarized in T able 9. W e ha ve a nice e xam ple here of what happens if we change t he order in which we deal with the sh ufﬂ es and the stuf ﬂes. W e reran the program of T able 9 for the weights w = 14 and w = 15 under these conditions, obtaining run ning times of 100973 and 493489 sec respectiv ely . This is more than an order of magnitude s lower than the order we s elect in the regular programs. Because we like to compare results of the M ZV runs wit h t hose of the Euler runs to as high a weight as possible we made also runs in which we do all calculus modulus a 31-bit prim e number . The number we selected is 2147479273. W e never ran into a case in which this seemed to cause problem s. In the program s in which we used this modulus we also dropped all terms t hat are products of lo wer weight objects . This means that in the end all sums are expressed into elements from the same-weigh t L yndon part o f the basis only . Such programs are much f aster . T his can be seen in T ables 10, 11 and 12 which are for depth d ≤ 4 , depth d ≤ 5 and depth d ≤ 6 , respective ly . The run at w = 18 , d = 6 deserves s ome special attention. It was our most costly run and during the running TFORM processed more than 7 · 10 12 terms. W e come now to our run s for the Multi ple Zeta V alues. Those runs look more spec- 32 weight constants running time [sec] output [Mbyte] 13 16812 388 5.5 15 33388 2932 18 17 60044 18836 53 19 100236 118874 131 21 157932 554870 299 T able 11: Summary of the runs at d = 5 in mo dular arith metic, dropping all terms that are products of lower weig ht objects. weight constants remaining running time [sec] output [Mbyte] 13 56940 22 2611 14 90564 37 12716 51 15 138636 35 55204 87 16 205412 66 206 951 214 17 295916 55 789 540 288 18 416004 109 2622157 711 T able 12: Summary of the runs at d = 6 in mo dular arith metic, dropping all terms that are products of lower weight objects. T imes refer to an 8 Xeon core machine at 3 GHz and 32 GBytes of memory . tacular because there i s much mo re l iterature on them. First we present the ‘complete’ runs i n whi ch all c alculus is over the ration al num bers and all terms are kept, cf. T able 13. ‘Rat’ i s the real time of this run divided by the real time of a run with a 31-bit prime number dropping also p roducts of lower weight obj ects. T ogether with the numbers in the ‘num’ column it shows that maki ng s e veral runs mod ulus a 31-bi t p rime and then using the Chinese remainder theorem [53], will not be ef ﬁcient. W e would need at least 12 runs for the w = 22 case and even then we ha ve to account for dropping the lo wer weight terms. W e indicate the m aximum value of the depth which, du e to the duality relation for MZVs, is sufﬁ cient to obtain all MZVs at the giv en w eight. The basis in which these resul ts are presented is described in Appendix B. If we let the program select the basis, the outputs are shorter b u t from the vie w point of basis e lements selected there is less structure. The ne xt sequence of runs is performed usin g in modular arithmetic in which we r efer to the same 31-bit prime number as before. Again we run the full range of depths needed to obtain all sums. As usual in modul ar runs , we drop the products of lower weight objects. The results are given i n T able 14. The output of the run at w = 23 giv es the results for 2 20 MZVs expressed in terms of the 28 same-weight elements of a L yn don basis selected by the program. In T able 15 we give the s tatistics of runs to a more restricted depth. If th e conjecture [13] is correct the runs at w = 25 , 26 should still gi ve us a complete basis. In the higher runs some elements will be missin g. W e would hav e liked t o ha ve a run for depth d ≤ 9 at w = 2 7 , but i t would probably 33 w d G s ize output num CPU[sec] real[sec] Eff . Rat. 16 8 128 1 1M 7M 22 2 89 56 5.16 0.99 17 8 256 3 0M 21M 19 6 77 1 29 5.25 0.96 18 9 256 8 8M 64M 29 3071 517 5.9 4 1.11 19 9 512 224M 182M 28 6848 1206 5.68 1.0 0 20 10 512 790M 558M 36 44883 6834 6.57 1.4 2 21 10 1024 1766M 1821M 40 86318 13851 6.23 1.12 22 11 1024 8856M 5927M 46 1572605 208972 7.53 3.1 8 T able 13: Runs on an 8-core Xeon computer at 3 GHz and with 32 Gbytes of memory . ‘Num’ indicates, for the ﬁnal expressions, the maxim um numb er of decimal digits in either a num erator or a denominator . ‘Eff. ’ is the ratio of CPU time versus real time indicating how well the processors are used. The meaning of the c olumn labeled ‘Rat. ’ is explained in the text. The anomaly between si ze and outp ut for w = 21 is due to the fact that the output is in text and size is in FORM binary notation. w G size output CPU[sec] real[sec] Eff. 16 128 1.7M 1.2M 300 5 7 5.25 17 256 5.6M 3.2M 713 134 5 .32 18 256 14 .4M 7.2M 2706 465 5.82 19 512 39M 19M 6901 1206 5.72 20 512 104M 4 5M 30097 4819 6.25 21 1024 239M 114M 75302 12379 6.08 22 1024 767M 280M 44920 2 65644 6.8 4 23 2048 2.17G 734M 992431 15 1337 6.56 24 2048 8.04G 1.77G 9251325 126 8247 7.29 T able 14: Runs on an 8-core Xeon computer at 3 GHz and with 32 Gbytes of memory . G is the size of the group used in the Gaussian eli mination, ‘size’ is the maximum s ize of the master expression during the run, ‘output’ is the size of t he mast er expression in the end, CPU is the tot al CPU time of all processors together in seconds, ‘real’ denotes the elaps ed ti me in seconds and ‘Eff. ’ is the pseudo efﬁcienc y , deﬁned b y the CPU tim e divided by the real time. 34 w D G size output CPU[sec ] real[sec] Eff. 23 7 2048 1.55G 89M 6144 7 95 79 6.41 24 8 2048 673M 380M 536921 72991 7.36 25 7 4096 6.37G 244M 369961 501 97 7.37 26 8 4096 38.3G 1160M 4786841 6515 39 7.35 27 7 6144 12.7G 914M 2152321 277135 7.77 28 6 6144 2.88G 314M 235972 309 60 7.62 29 7 6144 41.0G 3007M 8580364 1112836 7.71 30 6 6144 6.27G 658M 829701 10 6353 7.80 T able 15: Runs on an 8-core Xeon computer at 3 GHz and wi th 32 Gb ytes of mem- ory . D indicates the maximum depth (see text). W e reran at w = 23 and w = 24 t o ha ve information for extrapolation purposes. take more than a year with current technology . A run for depth d ≤ 8 at w = 28 will require a sm aller CPU time. The reason why t hese runs are interestin g is explained in Section 10 on pushdowns. They may giv e us a new typ e of basis element that would indicate a double pushdown. The outputs of all of t he abov e runs are collected in the data mine, t ogether with some ﬁles in which the results hav e been processed to make them more accessible. At t he end of this Section we would li ke to discuss the statu s of the general in vesti- gation of MZVs and Euler sums in the fore going literature. The relations between MZVs were s tudied bot h by mathematici ans and physicists. A n early study is due to Gastm ans and Troost [54], which g a ve a nearly com plete list for the Euler su ms of w = 4 and many relations for w = 5 , supplem ented in [11] later . V ariou s authors, among them D. Broad- hurst, to w = 9 , and D. Zagier , performed precisi on numerical studi es [55] using PARI [56] during the 1990’ s for MZVs, which were not published. A very far-r eaching in vestiga- tion concerned t he study of some of the MZVs at w = 23 and depth d = 7 by Broad- hurst by numerical techniques (PSLQ). Double sums were studied in [57] using the PSLQ method [15]. V ermaseren both s tudied the MZVs and th e Euler sums to w = 9 [10] using a FORM program [21]. This was the sit uation around the year 2000, when the Lille group presented their w = 12 results for the MZVs and w = 7 results for th e Euler sums [58 ]. In Ref. [59] the solution of w = 8 for the Euler sums is mentioned by the Lille–group. How- e ver , the data-tables made av ailable [58] only contain th e relations to w = 7 . Moreover , the relatio ns used in [59] do not cover the doubli ng relation, which is needed to reduce to the con jectured basis at this weigh t, as wil l be shown later . For the MZVs w = 10 had been solved i n [60] and w = 13 in [61], cf. [62]. V ermaseren could extend the MZVs to w = 16 [63]. Studies for w = 16 w ere also performed at Lill e [64] without making the results public. In the s tudies by V ermasere n also th e diver gent harmonic sum s ζ 1 , ~ a were included, as this is sometimes necessary for physics applications, cf. also [11]. The prim ary goal in this paper is to derive explicit representation s of the M ZVs over sev eral bases suitable to t he respecti ve questions i n vestigated. If one on ly wants to de- termine the size of the basis one may proceed dif ferently , cf. [50]. Here for w = 19 in the M ZV case it was sh own, that t he basis has the e xpected length, but modul o po wers of π 2 at e ven weights. In [51] the case w = 20 was studied determining the size of the 35 basis calculating the rank of the associated matrix modulo a 15-bi t pri me. Althou gh t he computation tim es are not excessive, hi gher weights could not be i n vestigated yet because of memory limit ations. Since these methods are based on the respective algebra onl y th ey can be extended to colored multiple z eta values by e xtend ing the underlying alphabet. 7 The Data Mine The re sults of our runs, together with a num ber of FORM programs to manip ulate them and clarifying text, are a vailable on the internet in pages that we call the MZV dat a min e. It can be located as a l ink in the FORM home page [65]. Here we wil l describe the notat ions and how to use the programs. The notations we use in the data mine are t hat t he MZ Vs are represented either by a function Z of which the var iables are its indices or by a si ngle symbol that consists of a string of objects of which the ﬁrst character i s the letter z and the remaining characters are decimal digits. Each of th ese s trings refers t o an ind ex of the MZV . Let us give an example : z11z3z3 = Z(11,3,3) For t he Euler sum s we use mos tly the function H . It can have posit iv e and negative indices, the negative ones in dicating alternating o r Euler sums. When we use basis elements a compact notation is t he l etter h followed by a number of alphabetic characters or dig- its. Each character stands for a negati ve i ndex. The digits 1 , · · · , 9 s tand for the indices − 1 , · · · , − 9 and the upper case characters A , · · · , Z stand for the indices − 10 , · · · , − 35. W e had no need to go further in this notati on. The next e xample should illustrate this: hL33 = H(-21,-3 ,-3). If there is e ver an y doubt about which v ariable indicates which object one c an look in the corresponding library ﬁle (always included a s a ﬁ le with the extension .h in the direc tory in which the integrals reside) in the procedure ‘f rombasis’. For reasons of econom y 9 the H -functions with a single negative index have a different notation. They are related to the constants η k deﬁned by η k =  1 − 1 2 k − 1  ζ k . (7.1) In the program we call these constants e3,e5,... . In some cases we use a variable with a notatio n similar to the notation for t he MZVs, except for that the character z is replaced by the character a . aiajak = A(i,j, k) = Z(i,j,k) +Z(-i,j,-k )+Z(i,-j,- k)+Z(-i,-j,k) = H(i,j,k) -H(-i,j, k)-H(i,-j, k)+H(-i,-j,k ) 9 It turns o ut that the numb er of digits in the f ractions is som ewhat smaller in η - notation than in ζ - notation. 36 Here A is th e function deﬁned in (10.3). In e xceptional c ases we refer to Z -functions with n egati ve i ndices. The most com mon notation for th is in the literature is t o put a bar over t he numb er . Thi s is howe ver a notation t hat cannot be us ed in programs like FORM . Hence we use ne gativ e indices for the alternating s ums t here. For the sym bolic variables we us e the no tation for the MZVs but with the character m between z and the number : zm11zm3z3 = Z(-11,-3 ,3) = -H(-11,3,3 ) The programs run i n what we call i ntegral notation. This means that the master ex- pression has th e index ﬁelds of the functions E, H and HH 10 in terms of the three letter alphabet { 0 , 1 , − 1 } for Euler sums and t he two lett er alphabet { 0 , 1 } for MZVs. This is then the wa y the ou tputs are presented. Actually , internall y the whole string of i ndices is put togeth er as o ne lar g e ternary num ber for Euler sums and o ne lar g e binary numb er for MZVs. This speeds up the calculation, but m akes i t virtually impossible to interpret intermediate results. The out puts are p resented in a method that on e may con sider unu sual. In FORM it is often more efﬁcient to have one big expression, rather t han 2 20 expressions as would be the case for the MZVs at w = 23 . Hence the output contains function s H with the indices of the corresponding MZV and each H is m ultiplied by what th is MZV is equal to. In the case that we ﬁxed a basis this can be an expression that consist s o f symbols like we deﬁned abov e. In the case that we did the calculus mod ulus a prime and o nly wa nted t o determine a b asis, it wi ll be a n e xp ression that consists of terms that each contain a single function HH with its indices in integral notation. These HH functions form the basis. Often at the end of the program t here is a list of the HH functions used. Because FORM will print the output in such a way that the functions H are taken outside brackets, the contents of each brack et are what each H function is equal to. W ith a decent editor i t takes very fe w ( ≤ 4) edi t commands to con vert such output into the deﬁnition of 2 20 table elements. If this output should be u sed as input for o ther systems, th is can be do ne, provided that the expressions do not cause memory problems. The format is in principle compatib le with Pari/GP, Reduce and Maple . There may be a probl em with l ar ge coef ﬁcients. FORM does not like to make output lines that are longer than a typical screen width. Hence they are usual ly broken up after some 75 characters. This holds als o for long num bers. These are broken off by a backslash character and continued on the next line. Th e problem is usually th at FORM places some whi te space at the beginning of the line and s ome programs may ha ve problems w ith that. Hence one can use an editor to remove all white space (blanks and tabs) at the beginning of the lines. The data mine consists of sev eral parts. The main part is form ed by the d iffe rent data sets. The remainder ﬁles g iv e information about h ow to use th e data mine and li nks to other useful i nformation and/or programs. The data are divided ov er a number of directories, each containing the results of one type of runs for a range of v alues of the weight. In each directory there are sev eral types of ﬁles again. Th e log– ﬁles of the runs are sto red. These contain the run time statistics and the output of the runs in text form at. 10 The fun ction HH is th e same a s th e fu nction H . W e need two different names because w hen we presen t the results the function H marks the brackets and the function HH marks the remaining basis elements. 37 Then there are the table ﬁles. They are in text format and cont ain table deﬁnitions for FORM programs. Their extension is .prc as in mzv21.prc . Some o f these ﬁles ha ve been split into sev eral ﬁles because they become m uch to big to be handled con veniently . These tables can be read and com piled. Y et the case of th e M ZVs at w = 2 2 with its nearly 6 Gbytes can be too lar ge for a system with ‘on ly’ 16 Gbytes. If one do es not ha ve a bigger machin e to ones di sposal, one shou ld use either the binary .sav ﬁle or t he .tbl ﬁle deﬁned below . The third type of ﬁles are the binary .sav ﬁles. They can be used to read in the complete tables without having to go through the com piler and wit hout having to load the compl ete table as t able elements (which needs also big com piler buf fers). Finally we hav e created so-called tablebases which allo w very fast access to individual elements. A tablebase is a t ype of database for lar ge tables. They are particular to FORM and have been used with great success in a number of very large calculatio ns. Their working is explained in Ref. [66] and the FORM manual. The tablebase ﬁles ha ve traditionally t he .tbl extension. In each directory we have also the programs that were used to create the various ﬁles and in some cases some example programs. There is another section in t he data mine t hat cont ains pages in which it i s explained how to manipulate the information in the ﬁles. Although m any ﬁles are in text format it is not easy to manipul ate a 4 Gbyt e text ﬁle and hence it might become necessary to either use FORM and one of the binary ﬁles, or to use the STedi editor which has been us ed to manipulate th ese ﬁles on a comput er with 16 Gbytes of main memory . Links are set to these programs. FORM programs are provided for the most common manip ulations of t he data. They contain much comm entary . This should make it ea sy for th e user t o c ustomize the programs should the need arise. The data m ine is located at http://www .nikhef.nl / ∼ form/datam ine/datamine.html . Its structure is giv en in Figure 2 : 38 depth 3 W <= 29 depth 4 W <= 22 rational depth 5 W <= 17 alldepth W <= 12 Euler modular depth 4 W <= 30 basis depth 5 W <= 21 depth 6 W <= 18 datamine complete W <= 22 modular W <= 24 MZV limited W <= 30 programs other things Figure 2: Layout of t he data part of the data mine. In this ﬁgure we use the following names: complete Complete expressions ov er the rational numbers. modular Products of lower we ight terms are dropped and the computatio n is performed modulus a lar ge prime. limited As m odular b ut incomplete bases. rational Compl ete expressions o ver t he rational numbers. other things Con ventions, publications, help, links, etc. The main probl em with the data mi ne is it s size. Many ﬁles are seve ral Gbytes long . W e hav e used bzip2 on most ﬁles, because i t gives a better com pression ratio than gzip , ev en though it is m uch slower , bot h in compress ing and decompressi ng. But e ven wi th bzip2 the combined ﬁles are lar ger than 30 Gbytes. All prog rams are FORM (or TFORM ) codes. They will run with the latest versions of FORM (or TFORM ). The executables of FORM can be obtained from the FORM web site: http://www .nikhef.nl / ∼ form . Please remember the license condition : if you use FORM (or TFORM ) for a publication, you should refer to Ref. [21]. 39 8 FORM Aspects As mentioned the running of the p rograms used posed g reat challenges for FORM and TFORM . This is n ot simply a matter of whether the system contains errors. It is m uch more a matter of whether the system d eals with the problem in a sensible and efﬁcient way . Where are th e bott lenecks? What is inefﬁcient? A clear example is the con version between sum notation and integral notation. This can be programmed in one line: repeat id H(?a,n?!{- 1,0,1},?b) = H(?a,0,n- sig_(n),?b ); for going to integral notation and repeat id H(?a,0,n?! {0,0},?b) = H(?a,n+sig _(n),?b); for going to sum notat ion. It turns ou t t hat when one goes to lar ge weights (for in stance more than 20), this becomes very slow because i t in volves very much pattern matching. Considering also that the u se of harmonic sums is becoming more and more popular it was decided to b u ilt two ne w commands in FORM for this transformation: ArgImplode ,H; ArgExplode ,H; The ﬁrst one con verts H to sum notation and the second one to integral notation. Th is made the program noticeably faster and easier to read. Another additi on to FORM concerns built-in sh ufﬂ e and stuf ﬂe commands. One of the problems with sh ufﬂ es i s that the sim ple programm ing of i t usu ally gives many identical terms. This means that t he shufﬂe product of two MZVs can become very slow , which is illustrated by the following little prog ram: S n1,n2; CF H,HH; L F = H(3,5,3)*H (6,2,5); ArgExplode ,H; Multiply HH; repeat; id H H(?a)*H(n1 ?,?b)*H(n2 ?,?c) = +HH(?a,n1) *H(?b)*H(n 2,?c) +HH(?a,n2) *H(n1,?b)* H(?c); endrepeat; id HH(?a) *H(?b)*H(? c) = H (?a,?b,?c) ; .end Time = 37.38 sec Generated terms = 2496144 F Terms in output = 2146 Bytes used = 63176 By putt ing much combinatorics in t he built-in shuf ﬂe statement we could sol ve m ost of these problems (alth ough no t all as the combinato rics can become very complicated). W ith the shufﬂe command the program be comes: 40 S n1,n2; CF H,HH; L F = H(3,5,3)*H (6,2,5); ArgExplode ,H; Shuffle,H; .end Time = 0.0 1 sec Generated terms = 5163 F Terms in output = 2146 Bytes used = 63176 This is a great improvement of course. For the stufﬂ e product t hings are much easier . There we have the compl ication that there are two deﬁnitions. One is the product u sed for the Z -sums and the other is the product used for the S -sum s. W e ha ve resolved that by appending a + for the Z -notati on and a - for the S -notati on: stuffle,Z+ ; stuffle,S- ; Not only did this make the program signiﬁcantly fa ster , it also made it more readable. This wa y the s tufﬂe product of tw o Euler su ms in integral notation becomes in princi- ple (assuming that we are in integral notation): ArgImplode ,H; #call convertHt oZ(H,Z) Stuffle,Z+ ; #call convertZt oH(Z,H) ArgExplode ,H; except for that in the actual program we substit uted the contents of the two con version procedures. Of course for MZVs the con versions a re not needed and we can use just: ArgImplode ,H; Stuffle,H+ ; ArgExplode ,H; A th ird improvement concerns the parallelization. The original parallelization of TFORM [22] assumed the treatment of a single large expression of which the terms are distributed over the workers and l ater gathered i n by t he master . During the phase in which we ex ecute a Gaussian eliminati on insid e a grou p of id entities, thi s is very inef- ﬁcient, because we deal with m any sm all expressions, each giving a certain amount of overhea d when they a re distributed o ver the w orker threads. Hence it was decided to cre- ate a new form of parallelization in which the user tells the program that there are many small expressions coming. The reaction of the master thread is now to divide the expres- sions over the workers. It on ly has to tell each work er whi ch expression to do next, after which the work er is respon sible for obtaining its inpu t and writin g its ou tput. The o nly remaining inefﬁ ciencies are that the writing of the output causes a trafﬁ c jam because 41 that has to be done sequentiall y . The ﬁnal results are kept in principle in a single ﬁle or its cached version. Addi tionally , there may be som e load balancing problem i n the end. This load balancing becomes rapidl y less when the size of the groups of equation s that is treated becomes bigger . The runni ng of this phase of the program can g iv e n early ideal ef ﬁciencies. A fourth improvement concerns the fact that very lengt hy programs run a risk of dis- continuity . This could be a po wer failure or a sudden urge of t he service department to ‘update’ the system, etc. For this a f acilit y has been implemented inside FORM that allows one to make ‘snaps hots’ of the current in ternal state, cf. [67]. At a lat er mo ment one can then restart from the p oint of the snapsh ot. The completion of th is facility came howe ver too late to hav e a practical impact for this paper . The pos sibility to perform t he calculus modulus a prime nu mber has existed in FORM since its ﬁrst version. Much of it remained untested because these facilities had not been used extensi vely . It turned o ut to be necessary to redesign parts of it and add a fe w new features. Other aspects of TFORM performed amazingly well. W e ha ve seen the program running with eight w orkers who all eight had to enter the fourth stage of the sorting si multane- ously . This i s rather rare e ven for sing le threads and only h appens for very large expres- sions. It giv es a bit of a slow down due to the g reat am ount of dis k accesses, but it all worked without a ny problems. The most impressive single module result observed wa s Time = 15720.03 sec Generated terms =12026531 96013 FF Terms in output = 150844797 4 substituti on(7-sh)-7 621 By tes used = 36215 474400 The execution time is that of the m aster . Actually t he master spent 1000 CPU sec on this step and the eight worker s each almost 200000 CPU sec. One may wonder about th e probabili ty that calculations , done with a system un der dev elo pment, gi ve correct answers. W e ha ve se veral remarks concerning this topic: • Whene ver FORM failed, it wa s always in a very obvious way , like crashing because it couldn’t interpret something. • The full all-depth outpu ts from the MZVs up to w = 22 and the Euler s ums u p to w = 12 ha ve been tested numerically by completely i ndependent programs , run under PARI-GP [56]. • Because of both TFORM and the MZV programs being under de velopment many programs hav e been run at least se veral times with diffe rent conﬁgurations and/or diffe rent orders of solving the equations. • TFORM operates in a rather non-determini stic f ashion. T erms are rarely di stributed twice in the same w ay ov er th e work ers because the master s erves the workers when they have ﬁnished a task and this is usually not in the same order . In the case of errors this would lead to dif ferent result s in dif ferent runs. • There are ef fects that are e xpected on the basis of extrapolation, like the pushdowns and the construction of a basis. If anything goes wrong, such ef fects are absent. 42 • If for ins tance a term gets lost in a calcul ation over th e rational num bers, usually the output would ha ve terms with fractions that are abn ormally much more complicated than the others. Thi s is du e to the fact that in intermediate st ages the coefﬁcients are usually much m ore complicated t han at the end. Such terms are spo tted relatively easily . 9 Results Armed with the vast amount of information contained i n the data mine we start wi th having a look at a number o f conjectures in t his this ﬁeld. The y concern the number of basis elements, either just as a function of th e weight or as a functi on o f weight and depth. W e ﬁrst check some conjectures made in the literature using t he data min e and then describe the selection o f the basis to represent the Eu ler sums and MZVs in t he data mine. 9.1 Checkin g some Conjectur es with the Data Mine Zagier conjectur e [2]: The number of elements in a L yn don-basis for the MZVs at weight w is given by Eq. (A.13). ✷ As far as we can check, the Zagier conjecture holds to weight 22. Assu ming that in the modular calculus no term s were lost due to sp urious zeroes, we can say that it holds to weight 2 4. W ith t he additio nal assumptio n that all (L ynd on) basis elements have a depth of at m ost one third of the weight we can even say that it hol ds to wei ght 26. If we com- bine the ﬁndings in the thesis of Racinet [68] t hat there may be 2 basis elements of depth 9 for weight 27 with our runs to dept h 7, the Zagier c onjecture ho lds also at weight 27 . This conjecture is in accordance with th e upper bo und for the size o f the b asis b eing deri ved i n Refs. [14]. Hoffman conjectur e [69]: A F ibonacci-basis for the MZVs at a given weight w is f ormed out of MZVs the in dex set of which is formed out of all words o ver the alphabet { 2 , 3 } . ✷ W e coul d test the basis conjectured by Hoffman up to weight w = 22 . If we take the sub- var iety i n which we only look at the L yndon words made from the indi ces 2 and 3, we can e ven verify this L yndo n basis to weigh t 24. Because th is basis is n ot centered around the concept of depth, we cannot use the partial runs at larger weig hts and limi ted depths for further validation. Br o adhurst conjectu r e [12]: The number of basis element s of the Euler sums at ﬁxed weight w and depth d is gi ven by Eq. (3.4). ✷ All our run s for Euler sums are in com plete agreement wit h the Broadhurst conjecture about the size and the form of a basis for these sums . Thi s means complete veriﬁcation up to weight 12, for depth 6 veriﬁcation (in modular arith metic) to wei ght 18, for depth 43 5 complete veriﬁcation to weigh t 17 and mod ular veriﬁcation to weight 21. For d epth 4 these numbers are weight 22 and weight 30 respectiv ely . Br o adhurst-Kr eimer conjecture s [13]: The numb er of basis element s of the M ZVs at ﬁxed weight w and depth d is give n by Eq. (3.5). The number of basis elements for MZVs when expressed in terms of Euler sums in a minimal depth representation is giv en by Eq. (3.6 ) ✷ The runs for th e MZVs conﬁrm t his conjecture over a lar ge range, cf. T ables 16, 17. The second part of the conjecture is harder to check than the ﬁrst part, because for this we need the results for the corresponding Euler sums. Another conjectur e by Hoffman [3]: H 2 , 1 , 2 , 3 − H 2 , 2 , 2 , 2 − 2 H 2 , 3 , 3 = 0 (9.1) H 2 , 1 , 2 , 2 , 3 − H 2 , 2 , 2 , 2 , 2 − 2 H 2 , 2 , 3 , 3 = 0 (9.2) H 2 , 1 , 2 , 2 , 2 , 3 − H 2 , 2 , 2 , 2 , 2 , 2 − 2 H 2 , 2 , 2 , 3 , 3 = 0 (9.3) H 2 , 1 , 2 , 2 , 2 , 2 , 3 − H 2 , 2 , 2 , 2 , 2 , 2 , 2 − 2 H 2 , 2 , 2 , 2 , 3 , 3 = 0 (9.4) H 2 , 1 , { 2 } k , 3 − H { 2 } k + 3 − 2 H { 2 } k , 3 , 3 = 0 ✷ (9.5) W e veriﬁed these relations up to weight w = 22 . At w = 24 we checke d the weight-24 part, since we hav e only the modular representation at this le vel. There are identities for special patterns of indices as 2 ζ m , 1 = m ζ m + 1 − m − 2 ∑ k = 1 ζ m − k ζ k + 1 , 2 ≤ m ∈ Z , (9.6) cf. [1, 4] or ζ { 3 , 1 } n = 1 2 n + 1 ζ { 2 } n = 1 4 n ζ { 4 } n = 2 π 4 n ( 4 n + 2 ) ! , (9.7) conjectured in [2] and proven i n [47]. Another relation is ζ 2 , { 1 , 3 } n = 1 4 n n ∑ k = 0 ( − 1 ) k ζ { 4 } n − k ( ( 4 k + 1 ) ζ 4 k + 2 − 4 k ∑ j = 1 ζ 4 j − 1 ζ 4 k − 4 j + 3 ) (9.8) conjectured in [19] and proven in [70]. For t he Euler sums one ﬁnds, [71], ζ { 3 } n = 8 n ζ {− 2 , 1 } n . (9.9) In Ref. [19] conjectures were give n for special cases based on PSLQ , ζ { 4 , 1 , 1 } 2 = 3 π 4 16  ζ 6 , 2 − 4 ζ 5 ζ 3  − 41 π 6 5040  ζ 2 3 − 77023 π 6 14414400  + 397 8 ζ 9 ζ 3 + ζ 4 3 (9.10) ζ 2 , 2 , 1 , 2 , 3 , 2 = 75 π 2 32  ζ 8 , 2 − 2 ζ 7 ζ 3 + 34 225 ζ 2 5 + 4528801 π 10 61297236000  − 825 8 ζ 7 ζ 5 , (9.1 1) 44 w /d 1 2 3 4 5 6 7 8 9 10 1 2 1 3 1 4 5 1 6 0 7 1 8 1 9 1 0 10 1 11 1 1 12 1 1 13 1 2 14 2 1 15 1 2 1 16 2 3 17 1 4 2 18 2 5 1 19 1 5 5 20 3 7 3 21 1 6 9 1 22 3 1 1 7 23 1 8 15 4 24 3 1 6 14 1 25 1 10 23 11 26 4 2 0 27 5 27 1 11 36 23 2 28 4 2 7 45 16 29 1 14 50 48 7 30 4 3 5 73 37 2 T able 16: Number of basis elements for MZ Vs as a functi on of weight and depth in a minimal depth representation. Underli ned are t he values we have veriﬁed wit h our programs. 45 w/d 1 2 3 4 5 6 7 8 9 10 1 2 1 3 1 4 5 1 6 7 1 8 1 9 1 10 1 11 1 1 12 2 13 1 2 14 2 1 15 1 3 16 3 2 17 1 5 1 18 3 5 19 1 7 3 20 4 8 1 21 1 9 7 22 4 1 4 3 23 1 12 14 1 24 5 2 0 9 25 1 15 25 4 26 5 3 0 20 1 27 1 18 42 12 28 6 4 0 42 4 29 1 22 66 30 1 30 6 5 5 75 15 T able 17: Nu mber of basis elements for MZVs as a function of weight and depth when expressed as Eu ler sums in a m inimal depth representation. Underlined are the v alues we hav e veriﬁed with our programs. 46 which we veriﬁed. A series of special relations for the Euler sums were conjectured in [19] based on PSLQ : ζ 2 , 1 , − 2 , − 2 = 39 128 ζ 4 ζ 3 − 193 64 ζ 5 ζ 2 + 593 128 ζ 7 (9.12) ζ − 2 , − 2 , 1 , 2 = 9 128 ζ 4 ζ 3 + 447 128 ζ 5 ζ 2 − 1537 256 ζ 7 (9.13) ζ {− 3 , 1 } 2 = − 7  α 5 − 39 64 ζ 5 + 1 8 ζ 4 ln ( 2 )  ζ 3 +  2 α 4 − 1 4 ζ 4  2 + 2  α 4 − 15 16 ζ 4 + 7 8 ζ 3 ln ( 2 )  2 − 1 32 ζ 8 . (9.14) Here α n = L i n ( 1 / 2 ) + ( − 1 ) n " ln n ( 2 ) n! − ζ 2 2 ln ( n − 2 ) ( 2 ) ( n − 2 ) ! # . (9.15) These relations are veriﬁed analytically as well by our data base. Relations (9.10 – 9.13) were also obtained in [58]. In Ref. [12] a series of relatio ns was conjectured for weigh t w = 8 ...12 and d = 3,4 for Euler sums being related to va lues ζ −| a 1 | , −| a 2 | . ζ 3 , − 3 , − 3 = 6 ζ 5 , − 1 , − 3 + 6 ζ 3 , − 1 , − 5 − 315 32 ln ( 2 ) ζ 3 ζ 5 + 6 ζ − 5 , − 1 ζ 3 47 + 40005 128 ζ 2 ζ 7 − 39 64 ζ 3 3 + 1993 256 ζ 3 ζ 6 + 8295 128 ζ 4 ζ 5 − 226369 384 ζ 9 , (9.16) ζ 3 , − 5 , − 3 = 1059 80 ζ 5 , 3 , 3 + 15 ζ 7 , − 1 , − 3 + 15 ζ 3 , − 1 , − 7 + 701 69 ζ − 5 , − 3 ζ 3 + 15 ζ − 7 , − 1 ζ 3 − 6615 256 ln ( 2 ) ζ 3 ζ 7 − 11852967 2560 ζ 11 + 301599 128 ζ 2 ζ 9 − 124943 5888 ζ 2 3 ζ 5 + 1753577 35328 ζ 3 ζ 8 + 2960103 5120 ζ 4 ζ 7 + 3405 32 ζ 5 ζ 6 , (9.17) ζ 3 , − 1 , 3 , − 1 = 61 27 ζ − 3 , − 3 , − 1 , − 1 − 14 3 ζ − 5 , − 1 , − 1 , − 1 − 185 27 ζ − 5 , − 1 ζ 2 − 163499 22356 ζ − 5 , − 3 + 2051 54 ζ − 7 , − 1 + 28 9 ln 2 ( 2 ) ζ − 5 , − 1 + 35 96 ln 2 ( 2 ) ζ 2 3 − 581 64 ln 2 ( 2 ) ζ 6 − 8735 576 ln ( 2 ) ζ 2 ζ 5 − 903 64 ln ( 2 ) ζ 3 ζ 4 − 1441 288 ζ 2 ζ 2 3 + 10365875 476928 ζ 3 ζ 5 + 36916435 1907712 ζ 8 . (9.18) 2 5 · 3 3 ζ 4 , 4 , 2 , 2 = 2 5 · 3 2 ζ 4 3 + 2 6 · 3 3 · 5 · 13 ζ 9 ζ 3 + 2 6 · 3 3 · 7 · 13 ζ 7 ζ 5 + 2 7 · 3 5 ζ 7 ζ 3 ζ 2 + 2 6 · 3 5 ζ 2 5 ζ 2 − 2 6 · 3 3 · 5 · 7 ζ 5 ζ 4 ζ 3 − 2 8 · 3 2 ζ 6 ζ 2 3 − 13177 · 15991 691 ζ 12 + 2 4 · 3 3 · 5 · 7 ζ 6 , 2 ζ 4 − 2 7 · 3 3 ζ 8 , 2 ζ 2 − 2 6 · 3 2 · 11 2 ζ 10 , 2 + 2 14 ζ − 9 , − 3 . (9.19) These relations were veriﬁed using the current data base. Eq. (9.19) is particularly in- teresting s ince it i mplies a relation between MZVs m ediated by one term o f t he k ind ζ −| a 1 | , −| a 2 | . There is a series of Theorems prov en on t he MZVs, whi ch can be veriﬁed using the data base. W e used already the duality theorem [2]. For th e M ZVs a large v ariety of relations has been proven, which can be veriﬁed for speciﬁc examples using the d ata mine. The ﬁrst of these general relations is the Sum Theor em , Ref. [1, 72], ∑ i 1 + ... + i k = n , i 1 > 1 ζ i 1 ,..., i k = ζ n . (9.20) The sum-theorem was conjectured in [40], cf. [39]. For its deri v ation us ing the Euler connection formula for polylogarit hms, cf. [73]. Further identi ties are given by t he Derivation Theorem , [40, 52] Let I = ( i 1 , . . . , i k ) any sequence of positi ve int egers with i 1 > 1 . Its deriv ati on D ( I ) is given by D ( I ) = ( i 1 + 1 , i 2 , . . . , i k ) + ( i 1 , i 2 + 1 , . . . , i k ) + . . . ( i 1 , i 2 , . . . , i k + 1 ) ζ D ( I ) = ζ ( i 1 + 1 , i 2 ,..., i k ) + . . . + ζ ( i 1 , i 2 ,..., i k + 1 ) . (9.21) The Deriv ation Theorem states ζ D ( I ) = ζ τ ( D ( τ ( I ))) . (9.22) 48 Here τ denotes the duali ty-operation (2.23). W e call an index-word w admiss ible, if its ﬁrst letter i s not 1 . The words form the set H 0 . | w | = w is the weight and d ( w ) the depth of w . For the MZVs the words w are b uild in terms of concatenation prod- ucts x i 1 − 1 0 x 1 x i 2 − 1 0 x 0 ... x i k − 1 0 x 1 . The heig ht of a word, h t ( w ) , counts the nu mber of (non- commutative) factors x a 0 x b 1 of w . The opera tor D and i ts dual D act as follows [7], Dx 0 = 0 , Dx 1 = x 0 x 1 , Dx 0 = x 0 x 1 , Dx 1 = 0 . Deﬁne an anti-symm etric deri vation ∂ n x 0 = x 0 ( x 0 + x 1 ) n − 1 x 1 . A generalization of the Deriv ation Theorem was gi ven in [52, 74] : The identity ζ ( ∂ n w ) = 0 (9.23) holds for any n ≥ 1 and any word w ∈ H 0 . Further theorems are t he Le–Murakami Theo- r em , [75], t he Ohno Theor em , [76], w hich generalizes the sum- and duality t heorem, t he Ohno–Zagier Theor em , [77], which covers the Le–Murakami theorem and the s um the- orem, and generalizes a theorem by H of fman [39, 40], and the cyclic sum theorem , [78]. Finally , we menti on a main conjecture for the MZVs. Consider t uples k = ( k 1 , . . . , k r ) ∈ N r , k 1 ≥ 1. One deﬁnes Z 0 : = Q Z 1 : = { 0 } Z w : = ∑ | k | = w Q · ζ ( k ) ⊂ R . (9.24) If further Z Go : = ∞ ∑ w = 0 Z w ⊂ R ( Gon charov ) (9.25) Z Ca : = ∞ M w = 0 Z w ( Cartier ) (9.26) the conjecture states (a) Z Go ∼ = Z Ca . There are no relations over Q between the MZVs of diff erent weight w . (b) dim Z w = d w , with d 0 = 1 , d 1 = 0 , d 2 = 1 , d w = d w − 2 + d w − 3 . (c) Al l relations between MZ Vs are given by the extended doubl e-shufﬂ e relations [79], cf. also [80]. If this conjecture turns out to be true all MZVs are irrational numbers. 9.2 Selection of a Basis Thus far we have not speciﬁed which basis we hav e been using for the MZVs. In ﬁrst instance, we actuall y let th e program select the basis. The result was the collection of remaining elements after elimi nation of as many elements as po ssible. Th e ordering in 49 the eliminatio n process was such that the remaining elements would be mi nimal in depth and m aximal in their sum notation. Hence Z 20 , 2 , 1 , 1 would be prefe rred ov er Z 18 , 4 , 1 , 1 . As it turned out, all remaining elements had an index ﬁeld which formed a L ynd on w ord. This is n ot really surprising due to the orde ring. Unfortunately there was not much systematics found in these elements. Next came the idea that if the Euler sums h a ve a basis m ade out of L ynd on words of only negative odd indices, maybe o ne s hould in vestigate t o which extent one can write a basis for the MZVs in terms o f L yn don words with positive odd indices onl y . It turns out that a number of elements ca n be selected with odd-only indices, b ut it is not possible for the whole basis. A num ber of basis elements needs at least two e ven indices. Deﬁnition. L w is the set of L ynd on words made out of posit iv e odd-int eger i ndices, with no index i = 1 at giv en weight w . ✷ W e observed that T able 17 can be reproduced by basis elements with indices i n L w . As mentioned, this is not a basis for the MZVs, b u t if we write as m any elements of the basi s as possible as elements of the set L w , the remaining elem ents of the basis ha ve a depth that is at l east t wo greater than the elements that are remaini ng in t he L w set and need at least two ev en indices. Ad ditionally , it loo ks like that they can be writt en as an extended version of these remaining elements by adding two indices 1 at the end and subtracting one from the ﬁrst two indices as in Z 7 , 5 , 3 → Z 6 , 4 , 3 , 1 , 1 . (9.27) W e h a ve been able to const ruct bases wi th these properties all t he way up to weight w = 26 . The complete (non-unique) recipe for such bases is: 1. Construct the s et L w of all L yndon words of positive odd integers excluding one that add up to w . 2. Starting at lo west depth, write as many basis elements of the basis as possi ble in terms of elements of L w . Call the remain ing elements in L w at this depth R ( D ) W . 3. At the ne xt depth, tw o units lar ger than the pre vio us one, write again as man y basis elements of the basis as possible in terms of elements of L w and construct R ( D + 2 ) W . 4. Write the elements of the basis wit h depth D + 2 that could not be writt en as ele- ments of L w as 1-fold extended elements of R ( D ) W . 5. Write the elements of th e basis with depth D + 2 that could not be written as el- ements of L w or 1-fold extended elements of R ( D ) W as 2-fold extended elements of what remains of R ( D − 2 ) W , etc. 6. If we are not done yet, raise D by t wo and go ba ck to step 3. The concept of n -fold extension is deﬁned by subtracting one from the ﬁrst 2 n indices and adding 2 n in dices with the v alue one at the end of the index set. T o ill ustrate this we giv e two e xamples. First the basis at weight w = 12 : 50 L 12 : H 9 , 3 H 7 , 5 P 12 : H 9 , 3 H 6 , 4 , 1 , 1 and next the basis at weight w = 18 : L 18 : H 15 , 3 H 13 , 5 H 11 , 7 H 9 , 3 , 3 , 3 H 7 , 5 , 3 , 3 H 7 , 3 , 5 , 3 H 7 , 3 , 3 , 5 H 5 , 5 , 5 , 3 P 18 : H 15 , 3 H 13 , 5 H 10 , 6 , 1 , 1 H 9 , 3 , 3 , 3 H 6 , 4 , 3 , 3 , 1 , 1 H 7 , 3 , 5 , 3 H 7 , 3 , 3 , 5 H 5 , 5 , 5 , 3 From the basis at wei ght 18 it should be clear why we put so much ef fort in obtaining the results for the Euler sums at weight 18, depth 6. Because the constructi on does no t tell which elem ents of L w to s elect the results are not unique. In fact quite a few selections are not possible because of dependencies between the elements of L w . Hence the whole procedure requires a certain amount of experiment- ing before a good basis is found. In Appendix B we ha ve tried to ﬁnd a basis in which the elements t hat are taken from L w hav e the highest values when their index set is seen as a multi-dig it number . Because of reasons bein g explained in the next Section we call t hese bases ‘pushdown bases’. W e do not ha ve complete runs for the weig hts w = 27 and w = 28 . In t hese cases the elements wit h the greatest depth are missing. But we can go through the con struction as far as possi ble and make predictions about the missing element s. It turns out that for both these weights a 2-fold extension is needed. For wei ght w = 27 this would be for dept h 5 to depth 9 and fo r weight w = 28 for depth 4 to depth 8. This concept was not taken into account in the conjectures in Ref. [13]. Hence we formulate a new conjecture that not only s peciﬁes the number of elements for each weight and dept h but also ho w many elements need how m any e x tensions. Conjectur e 2. The number of bas is elem ents D ( w , d , p ) of MZ Vs wi th wei ght w , depth d , and pus hdown p is obtained from the generatin g function ∞ ∏ w = 3 ∞ ∏ d = 1 ∞ ∏ p = 0 ( 1 − x w y d z p ) D ( w , d , p ) = 1 − x 3 y 1 − x 2 + x 12 y 2 ( 1 − y 2 z ) ( 1 − x 4 )( 1 − x 6 ) (9.28) solving for the coef ﬁcients of the monomials x w y d z p . ✷ This form ula predicts the ﬁrst n -fold extension ( n > 1) at weight w = 12 n + 3 and it wil l be to depth d = 4 n + 1. The exception is the ﬁrst extension at weight 12. W e show this in T able 18. It is a great pi ty that with the resources that were at our dispos al we just could not get di rect access to a double extension or pushdown. Ext rapolating from the num bers in T able 15 ind icates com puter t imes of the order of half a year (for weight 28, depth 8) to more than a year (for weight 27, depth 9). 10 Pushdowns As m entioned in the previous Section, there are elements that as MZVs can only b e written with a certain depth, while, when written i n terms of Euler sums, can be written with a 51 w/d 1 2 3 4 5 6 7 8 9 10 1 2 1 3 1 4 5 1 6 0 7 1 8 1 9 1 0 10 1 11 1 1 12 1 0,1 13 1 2 14 2 1 15 1 2 0,1 16 2 2,1 17 1 4 1,1 18 2 4,1 0,1 19 1 5 3,2 20 3 6,1 1,2 21 1 6 6,3 0,1 22 3 10,1 3,4 23 1 8 11,4 1,3 24 3 14,2 8,6 0,1 25 1 10 18,5 4,7 26 4 19,1 16,11 1,4 27 1 11 29,7 11,12 0,1,1 28 4 25,2 31,14 4,11,1 29 1 14 42,8 25,23 1,5,1 30 4 33,2 52,21 14 ,22,1 0,1,1 T able 18 : Number o f basi s elements for MZVs as a funct ion of weight, depth and exten- sion(or pushdown). If there are sev eral numbers, separated by c ommas, the ﬁrst indicates the number of elements that came from L w , the second t he number of 1-fold extensions from depth d − 2, the t hird the number o f 2-fold extensions from depth d − 4, etc. A si ngle number refers to the elements of L w . 52 smaller depth. This ph enomenon is called pushdown. The simplest example occurs at weight w = 1 2 and can be looked up in the T ables for the Euler sums . It is Z 6 , 4 , 1 , 1 = − 2107648 15825 H − 11 , − 1 + 50048 9495 H − 9 , − 3 − 117568 237375 H − 7 , − 5 + 100352 1583 ζ 2 H − 9 , − 1 − 3584 1583 ζ 2 H − 7 , − 3 + 320 57 ζ 2 2 H − 7 , − 1 − 64 171 ζ 2 2 H − 5 , − 3 − 253512822078 6914 481025690578 125 ζ 6 2 + 69528448 427275 η 3 η 9 − 32 35 η 2 3 ζ 3 2 + 64 243 η 4 3 − 21236224 299187 η 7 η 3 ζ 2 − 11072 1425 η 5 η 3 ζ 2 2 + 696654848 4984875 η 5 η 7 − 11690624 356175 η 2 5 ζ 2 , (10.1) in wh ich we remind the reader that η n = H − n . The n ext equation i s at weight w = 15 and is already considerably leng thier . The rhs o f Z 6 , 4 , 3 , 1 , 1 contains 49 terms when writt en in this form and some of t he fractions consist of more than 100 decimal digits. The phenomenon of th ese pu shdowns seems t o be intimately connected wit h the dou bling and generalized doubling relations. W e hav e in vestigated this at the weight w = 1 2 system. This is the only system over which we have complete control, because we hav e the full results for all depths for all Euler sums up to t his weight. If we run this system without the use of the doubli ng and generalized d oubling relation s there are three m ore elements l eft in th e ‘basis’, see T able 5. T wo are of depth 4 and o ne is of depth 2. And additionall y there is no p ushdown. The element Z 6 , 4 , 1 , 1 = H 6 , 4 , 1 , 1 needs on e of these extra elements at dept h 4. If we use the doubling relations, but we do n ot use th e GDRs, there is only o ne extra element of depth 4, b ut the pushdo wn does take place. If we us e only the GDRs, there a re no remaining elements beyond the re g ular basis and the pushdown takes place. Unfortunately we cannot run this t est for other weig hts. Not u sing the GDRs means that we cannot run at restricted depth, due to the phenomenon of leakage. Of course it is rather adventurous to make the statement that doubling is at the origin of the pus hdowns, when we have only a single case, b ut there is more supporti ng evidence as we will see below . The way we h a ve presented the p ushdown in (10.1), althou gh correct, is not i ts mos t transparent form. One can rewrite i t to as many MZVs as p ossible and obtain a much simpler representation. One can, for inst ance, write H − 9 , − 3 = 1055 1024 " − Z 9 , 3 − 185874 5275 ζ 7 ζ 5 − 37332 1055 ζ 9 ζ 3 + 1024 26375 H − 7 , − 5 + 187392 5275 H − 11 , − 1 + 92649159488 23101203125 ζ 6 2 # . (10.2) Additionall y , we introduce a new functi on A as A n 1 , n 2 , ··· , n p − 1 , n p = ∑ ± sH ± n 1 , ± n 2 , ··· , ± n p − 1 , n p (10.3) in which the sum is over the 2 p − 1 possible s ign combi nations and s = − 1 if the num ber of minus signs inside H is odd and s = + 1 if this number is e ven as in A 7 , 5 , 3 = H 7 , 5 , 3 − H − 7 , 5 , 3 − H 7 , − 5 , 3 + H − 7 , − 5 , 3 . (10.4) 53 Notice t hat the last index is always positive. In terms of the Z -notation t he functi on A is the sum ov er all Z -sum s with an e ven number of ne gativ e indi ces, b ut the abso lute v alues of the indices are identical to the indices of the A -function. W e re write then H − 7 , − 5 = − 25 3 " − A 7 , 5 + 1295 2304 Z 9 , 3 + 461399 15360 ζ 7 ζ 5 + 3213 128 ζ 9 ζ 3 − 126 5 H − 11 , − 1 − 39238805939 12612600000 ζ 6 2 # , (10.5) and ﬁnally the result for the pushdown b ecomes: Z 6 , 4 , 1 , 1 = − 64 27 A 7 , 5 − 7967 1944 Z 9 , 3 + 1 12 ζ 4 3 + 11431 1296 ζ 7 ζ 5 − 799 72 ζ 9 ζ 3 + 3 ζ 2 Z 7 , 3 + 7 2 ζ 2 ζ 2 5 + 10 ζ 2 ζ 7 ζ 3 + 3 5 ζ 2 2 Z 5 , 3 − 1 5 ζ 2 2 ζ 5 ζ 3 − 18 35 ζ 3 2 ζ 2 3 − 5607853 6081075 ζ 6 2 , (10 .6) which is much simpler t han equation (10.1). W e s ee the s ame happening in the expression for Z 6 , 4 , 3 , 1 , 1 , Z 6 , 4 , 3 , 1 , 1 = + 1408 81 A 7 , 5 , 3 + 16663 11664 Z 9 , 3 , 3 + 150481 68040 Z 7 , 3 , 5 + 10 ζ 3 Z 6 , 4 , 1 , 1 + 162823 3888 ζ 3 Z 9 , 3 − 17 20 ζ 5 3 − 101437 38880 ζ 5 Z 7 , 3 − 1520827 38880 ζ 3 5 + 1903 120 ζ 7 Z 5 , 3 − 93619 1296 ζ 7 ζ 5 ζ 3 + 3601 48 ζ 9 ζ 2 3 − 20651486329 4082400 ζ 15 + 14 5 ζ 2 Z 5 , 5 , 3 − 2 ζ 2 Z 7 , 3 , 3 − 27 ζ 2 ζ 3 Z 7 , 3 − 21 2 ζ 2 ζ 5 Z 5 , 3 − 61 2 ζ 2 ζ 2 5 ζ 3 − 84 ζ 2 ζ 7 ζ 2 3 + 31753363 12960 ζ 2 ζ 13 − 4 ζ 2 2 Z 5 , 3 , 3 − 5 ζ 2 2 ζ 3 Z 5 , 3 + 9 2 ζ 2 2 ζ 5 ζ 2 3 + 979621 1701 ζ 2 2 ζ 11 + 186 35 ζ 3 2 ζ 3 3 − 490670609 3572100 ζ 3 2 ζ 9 − 1455253 283500 ζ 4 2 ζ 7 + 4049341 311850 ζ 5 2 ζ 5 + 12073102 1488375 ζ 6 2 ζ 3 . (10.7) In both relations there is only a s ingle object in the equati on that is not an M ZV : the function A . This means that w e can write this A -function alt ernativ ely as a combi nation of MZVs of which o ne has a depth d ′ = d + 2 . W e h a ve done that w ith A 7 , 5 to obt ain (10.7), see the fourth term in th e right hand side. The intriguing part about i t all is that this function A contains half of the terms on the right hand side of the doubling relation in equation ( 2.16). In terms of H -functions it are the t erms in which the last index is positive and in terms of Z -funct ions it are all terms with an ev en number of negative indices. W e ha ve been able to construct pushdown relations for all extended basis elements up to weight w = 21 and o ne for weight w = 22 . Some of these coul d be constructed directly from the data mine. The more difﬁcult ones are, howe ver , out side the range of the ﬁles in the data mine. There we could use the data mine as an aid in limit ing the 54 search with num erical algorithms li ke LLL or PSLQ . More details are gi ven in Appendix C . This search for pushdowns is not always as simple as the two e xamples we ga ve above. Sometimes there is mo re than on e pushd own at a given depth , and sometim es there are elements at a gi ven depth that s hould be pushed do w n, but there are also elements that remain at that depth. In the last case it is usually a l inear combination of the extended element(s) and the remaining element(s) that get(s) pus hed down. But for all cases that we could check there is a sin gle function A associated with each pushed do wn element . If there are s e veral pushdowns at a given weight and d epth the right hand side may contain linear combi nations of the corresponding A -functi ons. In all cases we could select the bases such that t he index ﬁelds o f the A -functions correspon ded to the index ﬁelds o f the elements of the set L w that had to be extended. The above indicates that th ese A -functions h a ve a special statu s within the Eul er sums . They are quite similar to the MZVs. It should b e noted t hat not all A -functions ca n be written in terms of MZVs only . This holds only for a limited subset as we will see in the next Section. Addit ionally , not all A -functions that can be rewritten in terms of M ZVs can b e used for pushdowns, because a number of them can be rewr itten in terms of MZVs that have a t most the same depth as the A -function itself. The above ob serva tions lead to the following conjecture: Conjectur e 3. At each weig ht w , there exists a set of L yndon words L w from which on e may construct a basis for M ZVs as follows. For each L ynd on word one chooses either th e as sociated Z value or the associated A value, with the number of A values chosen to agree with the Broadhurst-Kreimer conjectures. Linear combinati ons of these A va lues then provide the pushdowns for the extensions of Z values by a pair unit indices, as exempliﬁed in Appendix C. ✷ What the above says is that we can ﬁnd a good basis for the MZVs usin g the set L w , provided we borrow som e elem ents from the Eul er sums. In such terms the basis for weight w = 1 8 would look like L 18 : Z 15 , 3 Z 13 , 5 Z 11 , 7 Z 9 , 3 , 3 , 3 Z 7 , 5 , 3 , 3 Z 7 , 3 , 5 , 3 Z 7 , 3 , 3 , 5 Z 5 , 5 , 5 , 3 P 18 : Z 15 , 3 Z 13 , 5 A 11 , 7 Z 9 , 3 , 3 , 3 A 7 , 5 , 3 , 3 Z 7 , 3 , 5 , 3 Z 7 , 3 , 3 , 5 Z 5 , 5 , 5 , 3 11 Special Euler Sums The disco very of the A -functions brings up a ne w point. Which Euler sums c an be written as a l inear com bination of MZVs only? This i s of course a perfect qu estion for a system like the data mine in whi ch e xhaus tiv e searches are relatively cheap. At the same t ime w e ask of course t he question whi ch A -fun ctions can be written in terms of MZVs on ly . W e should disting uish two cases : • The object can be writt en in terms of M ZVs that hav e at m ost the same depth a s the object. 55 w/d 2 3 4 5 7 13 9 2 0 8 5 10 8 2 9 19 26 2 0 10 7 22 17 7 11 25 38 6 0 12 9 40 43 13 13 31 62 4 1 14 11 62 77 23 15 37 90 6 3 16 13 90 137 34 17 43 121 6 3 T able 19: Number of Eu ler sums with at least on e negativ e index that can be rewritten in terms of MZVs only as a function of weight ( w ) and depth ( d ) . w/d 2 3 4 5 7 4 5 2 0 8 5 8 4 0 9 6 13 9 3 10 7 18 17 7 11 8 25 31 17 12 9 32 49 34 13 10 41 74 67 14 11 50 106 116 15 12 61 148 192 16 13 72 198 298 17 14 85 259 449 T able 20: Num ber of A -funct ions that can b e rewritten in term s of MZVs on ly as a fun c- tion of weight ( w ) and depth ( d ) . • The object needs MZVs of a higher dept h. This occurs when there is already an A -function that is used in a pu shdown. In that case many other A -functions may be re written in terms of this A -function and MZVs of the same depth or lower dept h. W e ﬁnd that whenever the s econd case can occur , it will for a lar ge fraction of the A - functions of that depth. The n umber of H -functions with at least one negative inde x that can be rewritten completely i n terms of MZ Vs is given in T able 19. In T able 20 we show the same for the A -functions. Here there are clearly many more. Actually a sizable fraction of the A -functions can be rewritten like th is. For example, there are 1365 ﬁnite A -functions of w = 17 , d = 5 of which 449 can be re written in terms of MZVs only . Considering that a n umber of t he Euler sums can be re written in terms of MZVs onl y , one may raise the question wheth er t he push downs can be re writ ten in such a way that they do not have the A -functions, but rather hav e a sin gle Euler sum in their right hand side. This turned out to be a difﬁc ult question to answer , because the pushdown at w = 21 , d = 7 56 was very tim e con suming and took sever al d ays for each trial. At ﬁrst the nu mber of candidates wa s rather large. W e could make a list of candidates in a way , simil ar to that of T able 19 for w = 21 , d = 5 and see which Euler sums could be expressed in terms of MZVs and A 7 , 5 , 3 , 3 , 3 which is the object that was used in the pushdown 11 . Unfortun ately the results for w = 21 , d = 5 are in modul ar arithmetic and wit hout the products of lower weight objects. T rying s e veral elements of the list gave negati ve results indicating t hat many objects that give only MZVs for the terms with the same weight m ay ha ve terms that are products of Euler sums of a lower weight. Then, after con structing T able 19 we looked for patterns and we noticed that the only eli gible elements for w = 13 , w = 15 , w = 17 are Z 3 , − 2 , 3 , − 2 , 3 = H 3 , − 2 , − 3 , 2 , 3 Z 3 , − 4 , 3 , − 2 , 3 = H 3 , − 4 , − 3 , 2 , 3 Z 3 , − 2 , 3 , − 4 , 3 = H 3 , − 2 , − 3 , 4 , 3 Z 3 , − 6 , 3 , − 2 , 3 = H 3 , − 6 , − 3 , 2 , 3 Z 3 , − 4 , 3 , − 4 , 3 = H 3 , − 4 , − 3 , 4 , 3 Z 3 , − 2 , 3 , − 6 , 3 = H 3 , − 2 , − 3 , 6 , 3 . (11.1) T ryi ng to rewrite Z 3 , − 6 , 3 , − 6 , 3 in t erms of A 7 , 5 , 3 , 3 , 3 by m eans of LLL (a 1 30 elements search) gav e the desired resul t. Hence by no w all pushdowns have been obtained as well in terms of MZVs as in term s of one single Euler sum only . Unfortunately the index ﬁ eld of these Euler sums seems to be completely unrelated to the index ﬁ elds of our basis elements. 12 Outlook The data mi ne has gi ven us already much information and it may yield more yet. But the current results lea ve also many new questions. T o name a fe w: • Can the GDRs be deriv ed and/or written in a simp ler wa y? • Why can the GDRs resolve the problem of ‘leakage’ ? • Why do we need the doubling relations at all? • What is the relation between the doubling formula and the pushdowns? • Is it po ssible to see which A -functi ons can be used for pushdowns without needing the Euler sums of the data mine? • Can a pushdown basis be constructed without needing the MZVs of the data mine? In addition t here is som e ‘unﬁnis hed business’. W e did not g et mo re than partial e v- idence for t he double pushdowns at weight 27 and weight 28. Al though we can g uess the basi s at weight 27, an LLL search for t he com plete formula would in volve more than 800 elements and probably more than 10 times the number of digits than what our current 11 Originally we worked with A 9 , 3 , 3 , 3 , 3 and it was only at a very late stage that we co n verted to A 7 , 5 , 3 , 3 , 3 . Hence a numb er of the ‘raw’ results still refer to A 9 , 3 , 3 , 3 , 3 . 57 searches needed. Considering the asymptot ic beha v iour of the LLL algorithm, this would mean at least 10 7 times the computer time we neede d for the current determinations. The data mi ne approach is also not very attractive. There we would need the Euler sums to weight 27, dept h 9. This might need ev en more extra orders of magn itude in resources than for the LLL algorithm. What would be very welcome is an algorithm b y which we can determin e a (small) subs et of the Euler su ms that includes the A -functi ons and com- bine this subset with the M ZVs. For the MZV p art of these double pushdowns things look much brighter . In modular arithm etic the continuously improving hardware and software technology should place thos e runs wi thin reach soon. With a bett er ordering of the pro- cessing of th e equatio ns, which u nfortunately we do not ha ve, the runs could already be attempted. Again , ﬁnding non-trivial subsets to which o ne might limit oneself, would immediately lead to great progress as well. W e hope, that the empirical discoveries we made in t his paper for harmon ic sums up t o w = 30 will stimu late mathematical research and e ventually lead to proofs of more f ar reaching theorems in the future. Here we re- gard the consideration of the embedding of the MZVs into the Euler sums of im portance. Like wise one m ay consi der colored ‘M ZVs’ wit h ev en higher roo ts of unity [81] i n th e future, which ha ve not been the objectiv e of this paper . The data mine wil l be extended whene ver ne w and relev ant results ar e obtained. there is a h istory page that shows additions and corrections. If others ha ve interesting c ontribu- tions, they should contact one of the authors. Acknowledgmen ts . The work has been suppo rted in part by the r esearch prog ram of the Dutch F ou ndation for Fundamental Research of Matter (FOM), by DFG Sonderforschungsbereich Tra nsregio 9, Com putergest ¨ utzte Theoretische T eilchenphysik, the European Commi ssion MR TN HEPTOOLS u nder Cont ract No. MR TN-CT -2006-035505, J.V . would also like to thank the Hum boldt foundatio n for its generous su pport, and DESY , Zeuthen and t he University of Karlsruhe for it s hosp itality during thi s work. The runs for creating the data mine were done on the computer system of the Theoretical P article Physics group (TTP) at the Univ ersity of Karlsruhe a nd computers at DESY and Nikhef. 58 A Fibonacc i and L ynd on Ba ses at Fixed W eight In the past sev eral bases have been considered for both the MZVs and the Euler sums . In some of these the concept of depth i s not relev ant and hence for the counting rules we s hould sum over the depth. W e wil l dis cuss tho se bases in this Appendi x. For a number o f these bases conj ectures are formulat ed in the literature, which cannot be broken down ﬁxi ng the depth. The cou nting relation for t he MZVs was conjectured in [2, 13] and [12], respectiv ely . The v ector s pace o f MZVs c an be constructed allo wi ng basi s elem ents, which contain besides the ζ –values the in dex of which is a L yndo n word products of this t ype of ζ -v alues of lower weig ht. One basis of this kind is w = 2 ζ 2 (A.1) w = 3 ζ 3 (A.2) w = 4 ζ 2 2 (A.3) w = 5 ζ 5 , ζ 2 ζ 3 (A.4) w = 6 ζ 2 3 , ζ 3 2 (A.5) w = 7 ζ 7 , ζ 5 ζ 2 , ζ 3 ζ 2 2 (A.6) w = 8 ζ 5 , 3 , ζ 5 ζ 3 , ζ 2 3 ζ 2 , ζ 4 2 (A.7) w = 9 ζ 9 , ζ 7 ζ 2 , ζ 5 ζ 2 2 , ζ 3 3 , ζ 3 ζ 3 2 (A.8) w = 10 ζ 7 , 3 , ζ 5 , 3 ζ 2 , ζ 7 ζ 3 , ζ 2 5 , ζ 5 ζ 3 ζ 2 , ζ 2 3 ζ 2 2 , ζ 5 2 , etc . (A.9) The number o f these basis elements is coun ted by th e Padov an numbers, ˆ P k , [43], which hav e the same recursion as the Perrin numbers, b ut start from the initial v alues ˆ P 1 = ˆ P 2 = ˆ P 3 = 1. Their generating function is G ( ˆ P k , x ) = 1 + x 1 − x 2 − x 3 = ∞ ∑ k = 0 x k ˆ P k . (A.10) They also obey a Binet-li ke f ormula. The ﬁrst values are giv en in T able 21. w 1 2 3 4 5 6 7 8 9 10 ˆ P w 1 1 1 2 2 3 4 5 7 9 w 11 12 1 3 14 15 16 17 18 19 20 ˆ P w 12 16 2 1 28 37 49 65 86 1 14 151 w 21 22 2 3 24 25 26 27 28 29 30 ˆ P w 200 265 351 465 616 816 1081 1432 1 897 2513 T able 21: The ﬁrst 30 Pado v an numbers. The above basis is of th e Fibonacci type. Another basis of the Fibonacci type is the Hoff man basis [69] which consis ts of all elements of which the index ﬁeld is made up 59 w 1 2 3 4 5 6 7 8 9 10 P w 0 2 3 2 5 5 7 10 12 1 7 w 11 12 1 3 14 15 16 17 18 19 20 P w 22 29 3 9 51 68 90 119 158 209 277 w 21 22 2 3 24 25 26 27 28 29 30 P w 367 486 644 853 1130 1497 1983 2627 3480 4610 T able 22: The ﬁrst 30 Perrin numbers. from 2’ s and 3 ’ s only . If one uses the following construction it is easy to see that the number of basis elements follows the P adovan sequence. w = 1 / 0 w = 2 ( 2 ) w = 3 ( 3 ) . (A.11) The index words at weight w are give n by I w = ∪ | a | =( w − 2 ) ( 2 , I a ) ∪ ∪ | b | =( w − 3 ) ( 3 , I b ) . (A.12) Let us no w turn to L yndon bases for the MZVs. Using a W i tt-type relation [44] the size of the basis is conjectured to be giv en by l ( w ) = 1 w ∑ d | w µ  w d  P d , P 1 = 0 , P 2 = 2 , P 3 = 3 , P d = P d − 2 + P d − 3 , d ≥ 3 . (A.13) Here the sum runs over the divisors d of the weight w and P d denotes the Perrin- numbers [45, 46]. The y are giv en by the Binet-like formula P n = α n + β n + γ n , with α , β , γ the roots of x 3 − x − 1 = 0 (A.14) and can be deriv ed from the genera ting function G ( P k , x ) = 3 − x 2 1 − x 2 − x 3 = ∞ ∑ k = 0 x k P k . (A.15) The ﬁrst values are gi ven in T able 22. For the b asis different choices are possible, which yi eld equiva lent representations. Here we choose the basis i n terms of ζ –values, wi th an index ﬁeld which forms a L yn- don word. Our ﬁrst choice consist s of indices, which contain as widely as poss ible odd integers. In case of ev en weights in a series of cases also indices with only ev en numb ers occur from w = 12 onwards, as e.g. for w = 18 : ζ 15 , 3 , ζ 13 , 5 , ζ 9 , 3 , 3 , 3 , ζ 7 , 5 , 3 , 3 , ζ 5 , 5 , 5 , 3 , ζ 7 , 5 , 5 , 1 , ζ 8 , 2 , 2 , 2 , 2 , 2 , ζ 12 , 2 , 2 , 2 . (A.16) 60 w 1 2 3 4 5 6 7 8 9 10 l w 0 1 1 0 1 0 1 1 1 1 w 11 12 13 14 15 1 6 17 18 19 20 l w 2 2 3 3 4 5 7 8 11 13 w 21 22 23 24 25 2 6 27 28 29 30 l w 17 21 28 34 45 56 73 92 120 151 T able 23: Number of b asis elements of the L yndon b asis for the MZVs for ﬁxed weight w . A second natural choice is to take the afore mentioned Hof fman basis and select from it only t hose el ements of which the ind ex ﬁeld forms a L yndon word. Because the alge- braic relation s for the product of b asis elements of lower weight do n ot give objects t hat are closely related to the basis elements at the hig her weight, this basis is not used very frequently . As an example we consider the case w = 30 and calculate the s ize of the bases usi ng the W it t formula (A.13) resp. the num ber o f L ynd on words made up by the l etters 2 and 3 only with 2 < 3. 30 h as the following decomp osition 30 ≡ k i ∗ 3 + l i ∗ 2 = 2 ∗ 3 + 12 ∗ 2 = 4 ∗ 3 + 9 ∗ 2 = 6 ∗ 3 + 6 ∗ 2 = 8 ∗ 3 + 3 ∗ 2 . (A.17) W e n ow calculate the num ber of L yndon words for each of th ese contributions, with m i = k i + l i , n i = 1 m i ∑ d | m i µ ( d ) ( m i / d ) ! ( k i / d ) ! ( l i / d ) ! . (A.18) One obtains L { 2 , 3 } ( 30 ) = 1 14  14! 12!2! − 7! 6!  + 1 13 13! 9!4! + 1 12  12! 6! 2 − 6! 3! 2 − 4! 2! 2 + 2! 1! 2  + 1 11 11! 8!3! = 151 . (A.19) Using (A.13) the result is l ( 30 ) = 1 30 [ P 30 − P 15 − P 10 − P 6 + P 5 + P 3 + P 2 − P 0 ] = 1 30 [ 4610 − 68 − 17 − 5 + 5 + 3 + 2 − 0 ] = 151 . (A.20) A basis up to weight w = 17 for the MZVs was also constructed in [82]. For the Euler sums the Fibonacci basis is coun ted by the Fibonacci nu mbers. When we consider also all diver gent multipl e zeta values the Fibonacci sequence is m erely shifted. It is easily shown that the d iv er g ent Eul er sums can be represented by the con ver gent sum s and the element σ 0 . As in the MZV case we may span the vector space of the Euler sums 61 w 1 2 3 4 5 6 7 8 9 10 f w 1 1 2 3 5 8 1 3 21 3 4 55 w 11 12 13 14 15 16 17 18 19 20 f w 89 144 233 377 610 987 1597 2584 4181 67 65 T able 24: The ﬁrst 20 Fibonacci numbers. by formi ng a basis, whi ch includes products o f lower weight basis elem ents contain ed in a L yn don-basis. One basis of this type, used in the summer program [10] reads w = 1 ln ( 2 ) (A.21) w = 2 ζ 2 , ln 2 ( 2 ) (A.22) w = 3 ζ 3 , ζ 2 ln ( 2 ) , ln 3 ( 2 ) (A.23) w = 4 Li 4 ( 1 / 2 ) , ζ 3 ln ( 2 ) , ζ 2 2 , ζ 2 ln 2 ( 2 ) , ln 4 ( 2 ) (A.24) w = 5 Li 5 ( 1 / 2 ) , ζ 5 , Li 4 ( 1 / 2 ) ln ( 2 ) , ζ 3 ζ 2 , ζ 3 ln 2 ( 2 ) , ζ 2 ln 3 ( 2 ) , ζ 2 2 ln ( 2 ) , ln 5 ( 2 ) (A.25) w = 6 Li 6 ( 1 / 2 ) , ζ − 5 , − 1 , Li 5 ( 1 / 2 ) ln ( 2 ) , ζ 5 ln ( 2 ) , Li 4 ( 1 / 2 ) ζ 2 , Li 4 ( 1 / 2 ) ln 2 ( 2 ) , ζ 2 3 , ζ 3 ζ 2 ln ( 2 ) , ζ 3 ln 3 ( 2 ) , ζ 3 2 , ζ 2 2 ln 2 ( 2 ) , ζ 2 ln 4 ( 2 ) , ln 6 ( 2 ) , etc . (A.26) These bases are counted by the Fibonacci-numbers [42 , 83], f w + 1 , which obey t he same recursion relation as the Lucas numbers, but with the initial conditio ns f 0 = 0 , f 1 = 1. They are represented by the formula gi ven by J.P .M. Binet (1843) 12 f d = 1 √ 5   1 + √ 5 2 ! d − 1 − √ 5 2 ! d   , (A.27) and result from the generating function G ( f k , x ) = x 1 − x − x 2 = ∞ ∑ k = 0 x k f k . (A.28) The ﬁrst values are gi ven in T able 24. Another Fibonacci basis can be constructed as w = 0 / 0 w = 1 ( − 1 ) w = 2 ( 0 , − 1 ) . (A.29) H − 1 ( 1 ) and H 0 , − 1 ( 1 ) = H − 2 ( 1 ) are chosen as basis elements. Conjectur e 4. W ith the above starting conditions, cons ider the index w o rds at weight w to be I w = ∪ | a | =( w − 1 ) ( − 1 , I a ) ∪ ∪ | b | =( w − 2 ) ( − 2 , I b ) . (A.30) 12 The relation was known to Euler and Moivre. 62 The basi s elements for the Eu ler s ums are then given by the ζ -v alues with indices out of I w . The elements of which the index sets are a L yndo n word form a L yndon basis.  The Fibonacci ver sion of this basis seems to hav e been discovered independently by S. Zlobin, see Ref. [71]. This construction is analogous to that by Hoffman i n the case of M ZVs. It also uses a 2-letter alphabet. The dif ferent decompo sition of the weight w , ho we ver , leads to a basis of different l ength. Ag ain we may deri ve th e length of the basis us ing the W itt -formula (A.52) or count ing the basis elements as L yndon words of the index set (A.30 ). Let us giv e an e xample for w = 20 . 20 = k i ∗ 1 + l i ∗ 2 = 18 ∗ 1 + 1 ∗ 2 = 16 ∗ 1 + 2 ∗ 2 = 14 ∗ 1 + 3 ∗ 2 = 12 ∗ 1 + 4 ∗ 2 = 10 ∗ 1 + 5 ∗ 2 = 8 ∗ 1 + 6 ∗ 2 = 6 ∗ 1 + 7 ∗ 2 = 4 ∗ 1 + 8 ∗ 2 = 2 ∗ 1 + 9 ∗ 2 (A.31) Similar to the non-alternating case one obtains L {− 1 , − 2 } ( 20 ) = 1 19 19! 18!1! + 1 18  18! 16!2! − 9! 8!1!  + 1 17 17! 14!3! + 1 16  16! 12!4! − 9! 8!1!  + 1 15  15! 10!5! − 3! 2!1!  + 1 14  14! 8!6! − 7! 4!3!  + 1 13 13! 7!6! + 1 12  12! 8!4! − 6! 4!2!  + 1 11 11! 9!2! = 7 50 . (A.32) Like wise the W itt -formula (A.52) yields l ( 20 ) = 1 20 [ l 20 − l 10 − l 4 + l 2 ] = 1 20 [ 15127 − 123 − 7 + 3 ] = 750 . (A.33) The above basi s suf fers from t he same shortcoming as the Hoffman basi s in that the concept of depth lacks relev ance. Hence we di d not use it. In a similar way we can construct yet another Fibonacci basis: Conjectur e 5. W ith the starting condit ions of (A.29), consider the inde x words at weight w to be I w = ∪ | a | =( w − 1 ) ( − 1 , I a ) ∪ ∪ | b | =( w − 2 ) ( 0 , 0 , I b ) . (A.34) The basis elements for the Euler sums are then gi ven by th e ζ -values o f indices I w . The elements of which the index ﬁelds are a L yndon word and all i ndices are odd va lued i f w > 2 form a L yndon basis.  The L yndon basis of thi s construction happens to be the basis proposed in ref [12]. W e can divide I w I w = I odd w ⊕ I ¬ od d w , (A.35) 63 with the indices in I odd w are all odd and the last index of I ¬ od d w e ven, all others odd. The L y ndon words of I odd w , L y [ I odd w ] , form the basis elements at weight w and they are count ed by (A.52). Note, th at the basis element at w = 2 is not odd, which is an e xception. As an illust ration we consider t he case w = 6 . The following words are generated, where we assume the ordering 0 < 1 and let the digit 1 play the role of -1. { 000001 , 000011 , 00100 1 , 00 1101 , 001111 } ; { 100001 , 100101 , 10011 1 , 11 0001 , 110011 , 11 1001 , 111101 , 1 11111 } . (A.36) The L yndon words are ( 000011 ) ≡ ( − 5 , − 1 ) ; ( 0011 11 ) ≡ ( − 3 , − 1 , − 1 , − 1 ) ; ( 000001 ) ≡ ( − 6 ) ; ( 001101 ) ≡ ( − 3 , − 1 , − 2 ) . (A.37) The L yndon words with o dd indices taken as i ndex o f an Eul er sum are b asis elements, which we e xpress through the harmonic polylogarithm s at ar gument x = 1, H − 5 , − 1 ( 1 ) and H − 3 , − 1 , − 1 , − 1 ( 1 ) . On the ot her hand, H − 6 = 62 35 H 3 − 2 (A.38) H − 3 , − 1 , − 2 = H − 5 , − 1 + H − 2 H − 3 , − 1 + 452 105 H 3 − 2 − 55 18 H 2 − 3 (A.39) do not belong to the basis. The last L yndon basis is t he one we actually us e in the programs. It is dept h oriented and no element can be writ ten as a linear com bination of elements of lo wer depth or products o f elem ents wit h lower weight . T o weight w = 12 the complete basis for the ﬁnite elements is giv en by w = 1 H − 1 ; (A.40) w = 2 H − 2 ; (A.41) w = 3 H − 3 ; (A.42) w = 4 H − 3 , − 1 ; (A.43) w = 5 H − 5 , H − 3 , − 1 , − 1 ; (A.44) w = 6 H − 5 , − 1 , H − 3 , − 1 , − 1 , − 1 ; (A.45) w = 7 H − 7 , H − 5 , − 1 , − 1 , H − 3 , − 3 , − 1 , H − 3 , − 1 , − 1 , − 1 ; (A.46) w = 8 H − 7 , − 1 , H − 5 , − 3 , H − 5 , − 1 , − 1 , − 1 , H − 3 , − 3 , − 1 , − 1 , H − 3 , − 1 , − 1 , − 1 , − 1 ; (A.47) w = 9 H − 9 , H − 7 , − 1 , − 1 , H − 5 , − 3 , − 1 , H − 5 , − 1 , − 3 , H − 5 , − 1 , − 1 , − 1 , − 1 , H − 3 , − 3 , − 1 , − 1 , − 1 , H − 3 , − 1 , − 3 , − 1 , − 1 , H − 3 , − 1 , − 1 , − 1 , − 1 , − 1 , − 1 ; (A.48) w = 10 H − 9 , − 1 , H − 7 , − 3 , H − 7 , − 1 , − 1 , − 1 , H − 5 , − 3 , − 1 , − 1 , H − 5 , − 1 , − 3 , − 1 , H − 5 , − 1 , − 1 , − 3 , H − 3 , − 3 , − 1 , − 1 , H − 5 , − 1 , − 1 , − 1 , − 1 , − 1 , H − 3 , − 3 , − 1 , − 1 , − 1 , − 1 , H − 3 , − 1 , − 3 , − 1 , − 1 , − 1 , H − 3 , − 1 , − 1 , − 1 , − 1 , − 1 , − 1 , − 1 ; (A.49) 64 w 1 2 3 4 5 6 7 8 9 10 l w 1 3 4 7 11 18 29 47 76 123 w 11 12 13 14 15 16 1 7 18 19 2 0 l w 199 322 521 843 1364 2207 3571 5778 9349 15127 T able 25: The ﬁrst 20 Lucas numbers. w = 11 H − 11 , H − 9 , − 1 , − 1 , H − 7 , − 3 , − 1 , H − 7 , − 1 , − 3 , H − 5 , − 5 , − 1 , H − 5 , − 3 , − 3 , H − 3 , − 3 , − 1 , − 3 , − 1 , H − 3 , − 3 , − 3 , − 1 , − 1 , H − 5 , − 1 , − 1 , − 1 , − 3 , H − 5 , − 1 , − 1 , − 3 , − 1 , H − 5 , − 1 , − 3 , − 1 , − 1 , H − 5 , − 3 , − 1 , − 1 , − 1 , H − 7 , − 1 , − 1 , − 1 , − 1 , H − 3 , − 1 , − 1 , − 3 , − 1 , − 1 , − 1 , H − 3 , − 1 , − 3 , − 1 , − 1 , − 1 , − 1 , H − 3 , − 3 , − 1 , − 1 , − 1 , − 1 , − 1 , H − 5 , − 1 , − 1 , − 1 , − 1 , − 1 , − 1 , H − 3 , − 1 , − 1 , − 1 , − 1 , − 1 , − 1 , − 1 , − 1 ; (A.50) w = 12 H − 7 , − 5 , H − 9 , − 3 , H − 11 , − 1 , H − 5 , − 1 , − 3 , − 3 , H − 5 , − 3 , − 1 , − 3 , H − 5 , − 3 , − 3 , − 1 , H − 5 , − 5 , − 1 , − 1 , H − 7 , − 1 , − 1 , − 3 , H − 7 , − 1 , − 3 , − 1 , H − 7 , − 3 , − 1 , − 1 , H − 9 , − 1 , − 1 , − 1 , H − 3 , − 3 , − 1 , − 1 , − 3 , − 1 , H − 3 , − 3 , − 1 , − 3 , − 1 , − 1 , H − 3 , − 3 , − 3 , − 1 , − 1 , − 1 , H − 5 , − 1 , − 1 , − 1 , − 1 , − 3 , H − 5 , − 1 , − 1 , − 1 , − 3 , − 1 , H − 5 , − 1 , − 1 , − 3 , − 1 , − 1 , H − 5 , − 1 , − 3 , − 1 , − 1 , − 1 , H − 5 , − 3 , − 1 , − 1 , − 1 , − 1 , H − 7 , − 1 , − 1 , − 1 , − 1 , − 1 , H − 3 , − 1 , − 1 , − 3 , − 1 , − 1 , − 1 , − 1 , H − 3 , − 1 , − 3 , − 1 , − 1 , − 1 , − 1 , − 1 , H − 3 , − 3 , − 1 , − 1 , − 1 , − 1 , − 1 , − 1 , H − 5 , − 1 , − 1 , − 1 , − 1 , − 1 , − 1 , − 1 , H − 3 , − 1 , − 1 , − 1 , − 1 , − 1 , − 1 , − 1 , − 1 , − 1 ; (A.51) For the L yndon basis the conjectured length is [12] l ( w ) = 1 w ∑ d | w µ  w d  l d , w ≥ 2 l 1 = 1 , l 2 = 3 , l 3 = 4 , l d = l d − 1 + l d − 2 , d ≥ 4 . l ( 1 ) = 2 (A.52) l d denote the Lucas-numbers [46, 83]. They are represented by l d = 1 + √ 5 2 ! d + 1 − √ 5 2 ! d , (A.53) and deriv e from the genera ting function G ( l k , x ) = 2 − x 1 − x − x 2 = ∞ ∑ k = 0 x k l k . (A.54) The ﬁrst v alues are given in T able 25. The case w = 1 is special as two elements con- tribute. 65 w 1 2 3 4 5 6 7 8 9 10 l w 1 1 1 1 2 2 4 5 8 11 w 11 12 13 14 15 16 17 18 19 20 l w 18 25 40 58 90 135 210 3 16 492 750 T able 26: Number of basis elements of th e L yn don basi s for the Euler s ums for ﬁxed weight w . B Pushdown Bases W e ha ve tried to select a basis in which the el ements of the set L w are m aximal and the ex- tended elements are mi nimal. At the s ame time the extended elements shou ld be L yndon words. This means f or ins tance that a n element like H 5 , 5 , 5 , 3 cannot be e xtend ed and h ence has t o be part of the basis, even though it is t he minim al element at weight w = 18 . One could of course reve rse t he crit eria. For the const ruction of the bases thi s does not really diminish the amount of work. In both ca ses there are el ements that should be skipped be- cause of l inear dependencies. W e call the basis below the ‘minim al pus hdown basis’. In addition w e have used the requi rement t hat for the extended elem ents the corresponding A -function should be usable for a pushdown. This requirement we could enforce up to weight w = 22 . For higher weights we do not have the information in the data mi ne, and hence we do not know whether this requirement can be achie ved. P 2 = H 2 (B.1) P 3 = H 3 (B.2) P 5 = H 5 (B.3) P 7 = H 7 (B.4) P 8 = H 5 , 3 (B.5) P 9 = H 9 (B.6) P 10 = H 7 , 3 (B.7) P 11 = H 11 , H 5 , 3 , 3 (B.8) P 12 = H 9 , 3 , H 6 , 4 , 1 , 1 (B.9) P 13 = H 13 , H 7 , 3 , 3 , H 5 , 5 , 3 (B.10) P 14 = H 11 , 3 , H 9 , 5 , H 5 , 3 , 3 , 3 (B.11) P 15 = H 15 , H 7 , 3 , 5 , H 9 , 3 , 3 , H 6 , 4 , 3 , 1 , 1 (B.12) P 16 = H 11 , 5 , H 13 , 3 , H 5 , 5 , 3 , 3 , H 7 , 3 , 3 , 3 , H 8 , 6 , 1 , 1 (B.13) P 17 = H 17 , H 7 , 5 , 5 , H 9 , 3 , 5 , H 9 , 5 , 3 , H 11 , 3 , 3 , H 5 , 3 , 3 , 3 , 3 , H 6 , 6 , 3 , 1 , 1 (B.14) P 18 = H 13 , 5 , H 15 , 3 , H 5 , 5 , 5 , 3 , H 7 , 3 , 3 , 5 , , H 7 , 3 , 5 , 3 , H 9 , 3 , 3 , 3 , H 10 , 6 , 1 , 1 , H 6 , 4 , 3 , 3 , 1 , 1 (B.15) P 19 = H 19 , H 9 , 3 , 7 , H 9 , 5 , 5 , H 11 , 3 , 5 , H 11 , 5 , 3 , H 13 , 3 , 3 , H 5 , 3 , 5 , 3 , 3 , H 5 , 5 , 3 , 3 , 3 , H 7 , 3 , 3 , 3 , 3 , H 6 , 6 , 5 , 1 , 1 , H 8 , 6 , 3 , 1 , 1 (B.16) 66 P 20 = H 13 , 7 , H 15 , 5 , H 17 , 3 , H 7 , 3 , 5 , 5 , H 7 , 5 , 5 , 3 , H 7 , 7 , 3 , 3 , H 9 , 3 , 3 , 5 , H 9 , 3 , 5 , 3 , H 11 , 3 , 3 , 3 , H 10 , 8 , 1 , 1 , H 5 , 3 , 3 , 3 , 3 , 3 , H 6 , 4 , 3 , 5 , 1 , 1 , H 8 , 4 , 3 , 3 , 1 , 1 (B.17) P 21 = H 21 , H 9 , 5 , 7 , H 9 , 9 , 3 , H 11 , 3 , 7 , H 13 , 3 , 5 , H 13 , 5 , 3 , H 15 , 3 , 3 , H 5 , 5 , 3 , 5 , 3 , H 5 , 5 , 5 , 3 , 3 , H 7 , 3 , 3 , 3 , 5 , H 7 , 3 , 3 , 5 , 3 , H 7 , 3 , 5 , 3 , 3 , H 9 , 3 , 3 , 3 , 3 , H 8 , 6 , 5 , 1 , 1 , H 10 , 4 , 5 , 1 , 1 , H 10 , 6 , 3 , 1 , 1 , H 6 , 4 , 3 , 3 , 3 , 1 , 1 (B.18) P 22 = H 15 , 7 , H 17 , 5 , H 19 , 3 , H 7 , 5 , 7 , 3 , H 7 , 7 , 3 , 5 , H 9 , 3 , 5 , 5 , H 9 , 3 , 7 , 3 , H 9 , 5 , 3 , 5 , H 9 , 5 , 5 , 3 , H 11 , 3 , 3 , 5 , H 11 , 3 , 5 , 3 , H 11 , 5 , 3 , 3 , H 13 , 3 , 3 , 3 , H 12 , 8 , 1 , 1 , H 5 , 3 , 5 , 3 , 3 , 3 , H 5 , 5 , 3 , 3 , 3 , 3 , H 7 , 3 , 3 , 3 , 3 , 3 H 6 , 4 , 5 , 5 , 1 , 1 , H 6 , 6 , 5 , 3 , 1 , 1 , H 8 , 2 , 3 , 7 , 1 , 1 , H 8 , 6 , 3 , 3 , 1 , 1 (B.19) P 23 = H 23 , H 11 , 7 , 5 , H 11 , 9 , 3 , H 13 , 3 , 7 , H 13 , 5 , 5 , H 13 , 7 , 3 , H 15 , 3 , 5 , H 15 , 5 , 3 , H 17 , 3 , 3 , H 5 , 5 , 5 , 5 , 3 , H 7 , 3 , 7 , 3 , 3 , H 7 , 3 , 5 , 5 , 3 , H 7 , 5 , 3 , 5 , 3 , H 7 , 5 , 5 , 3 , 3 , H 7 , 7 , 3 , 3 , 3 , H 9 , 3 , 3 , 3 , 5 , H 9 , 3 , 3 , 5 , 3 , H 9 , 3 , 5 , 3 , 3 , H 9 , 5 , 3 , 3 , 3 , H 11 , 3 , 3 , 3 , 3 , H 8 , 6 , 7 , 1 , 1 , H 8 , 8 , 5 , 1 , 1 , H 10 , 2 , 9 , 1 , 1 , H 10 , 4 , 7 , 1 , 1 , H 5 , 3 , 3 , 3 , 3 , 3 , 3 H 6 , 2 , 3 , 5 , 5 , 1 , 1 , H 6 , 2 , 5 , 3 , 5 , 1 , 1 , H 6 , 4 , 3 , 3 , 5 , 1 , 1 (B.20) P 24 = H 17 , 7 , H 19 , 5 , H 21 , 3 , H 7 , 7 , 7 , 3 , H 9 , 7 , 3 , 5 , H 9 , 7 , 5 , 3 , H 9 , 9 , 3 , 3 , H 11 , 3 , 3 , 7 , H 11 , 3 , 5 , 5 , H 11 , 3 , 7 , 3 , H 11 , 5 , 3 , 5 , H 11 , 5 , 5 , 3 , H 11 , 7 , 3 , 3 , H 13 , 3 , 3 , 5 , H 13 , 3 , 5 , 3 , H 13 , 5 , 3 , 3 , H 15 , 3 , 3 , 3 , H 12 , 10 , 1 , 1 , H 14 , 8 , 1 , 1 , H 5 , 5 , 3 , 3 , 5 , 3 , H 5 , 5 , 3 , 5 , 3 , 3 , H 5 , 5 , 5 , 3 , 3 , 3 , H 7 , 3 , 3 , 3 , 5 , 3 , H 7 , 3 , 3 , 5 , 3 , 3 , H 7 , 3 , 5 , 3 , 3 , 3 , H 7 , 5 , 3 , 3 , 3 , 3 , H 9 , 3 , 3 , 3 , 3 , 3 , H 6 , 6 , 5 , 5 , 1 , 1 , H 8 , 2 , 5 , 7 , 1 , 1 , H 8 , 2 , 7 , 5 , 1 , 1 , H 8 , 4 , 3 , 7 , 1 , 1 , H 8 , 4 , 5 , 5 , 1 , 1 , H 8 , 4 , 7 , 3 , 1 , 1 , H 6 , 2 , 3 , 3 , 3 , 5 , 1 , 1 (B.21) P 25 = H 25 , H 11 , 11 , 3 , H 13 , 5 , 7 , H 13 , 7 , 5 , H 13 , 9 , 3 , H 15 , 3 , 7 , H 15 , 5 , 5 , H 15 , 7 , 3 , H 17 , 3 , 5 , H 17 , 5 , 3 , H 19 , 3 , 3 , H 7 , 3 , 7 , 3 , 5 , H 7 , 5 , 3 , 7 , 3 , H 7 , 5 , 7 , 3 , 3 , H 9 , 3 , 3 , 3 , 7 , H 9 , 3 , 3 , 5 , 5 , H 9 , 3 , 3 , 7 , 3 , H 9 , 3 , 5 , 3 , 5 , H 9 , 3 , 5 , 5 , 3 , H 9 , 3 , 7 , 3 , 3 , H 9 , 5 , 3 , 3 , 5 , H 9 , 5 , 3 , 5 , 3 , H 9 , 5 , 5 , 3 , 3 , H 9 , 7 , 3 , 3 , 3 , H 11 , 3 , 3 , 3 , 5 , H 11 , 3 , 3 , 5 , 3 , H 11 , 3 , 5 , 3 , 3 , H 11 , 5 , 3 , 3 , 3 , H 13 , 3 , 3 , 3 , 3 , H 8 , 8 , 7 , 1 , 1 , H 10 , 4 , 9 , 1 , 1 , H 10 , 6 , 7 , 1 , 1 , H 10 , 8 , 5 , 1 , 1 , H 12 , 2 , 9 , 1 , 1 , H 5 , 3 , 3 , 5 , 3 , 3 , 3 , H 5 , 3 , 5 , 3 , 3 , 3 , 3 , H 5 , 5 , 3 , 3 , 3 , 3 , 3 , H 7 , 3 , 3 , 3 , 3 , 3 , 3 , H 6 , 2 , 5 , 5 , 5 , 1 , 1 , H 6 , 4 , 3 , 5 , 5 , 1 , 1 , H 6 , 4 , 5 , 3 , 5 , 1 , 1 , H 6 , 4 , 5 , 5 , 3 , 1 , 1 , H 6 , 6 , 3 , 3 , 5 , 1 , 1 , H 6 , 6 , 3 , 5 , 3 , 1 , 1 , H 6 , 6 , 5 , 3 , 3 , 1 , 1 (B.22) P 26 = H 17 , 9 , H 19 , 7 , H 21 , 5 , H 23 , 3 , H 7 , 7 , 7 , 5 , H 9 , 5 , 9 , 3 , H 11 , 3 , 9 , 3 , H 11 , 5 , 3 , 7 , H 11 , 5 , 5 , 5 , H 11 , 5 , 7 , 3 , H 11 , 7 , 3 , 5 , H 11 , 7 , 5 , 3 , H 11 , 9 , 3 , 3 , H 13 , 3 , 3 , 7 , H 13 , 3 , 5 , 5 , H 13 , 3 , 7 , 3 , H 13 , 5 , 3 , 5 , H 13 , 5 , 5 , 3 , H 13 , 7 , 3 , 3 , H 15 , 3 , 3 , 5 , H 15 , 3 , 5 , 3 , H 15 , 5 , 3 , 3 , H 17 , 3 , 3 , 3 , H 14 , 10 , 1 , 1 , H 5 , 5 , 5 , 3 , 5 , 3 , H 5 , 5 , 5 , 5 , 3 , 3 , H 7 , 3 , 3 , 5 , 5 , 3 , H 7 , 3 , 5 , 3 , 5 , 3 , H 7 , 3 , 5 , 5 , 3 , 3 , H 7 , 3 , 7 , 3 , 3 , 3 , H 7 , 5 , 3 , 3 , 5 , 3 , H 7 , 5 , 3 , 5 , 3 , 3 , H 7 , 5 , 5 , 3 , 3 , 3 , H 7 , 7 , 3 , 3 , 3 , 3 , H 9 , 3 , 3 , 3 , 3 , 5 , H 9 , 3 , 3 , 3 , 5 , 3 , H 9 , 3 , 3 , 5 , 3 , 3 , H 9 , 3 , 5 , 3 , 3 , 3 , H 9 , 5 , 3 , 3 , 3 , 3 , H 11 , 3 , 3 , 3 , 3 , 3 , H 8 , 2 , 7 , 7 , 1 , 1 , H 8 , 4 , 5 , 7 , 1 , 1 , H 8 , 4 , 7 , 5 , 1 , 1 , H 8 , 6 , 3 , 7 , 1 , 1 , H 8 , 6 , 5 , 5 , 1 , 1 , H 8 , 6 , 7 , 3 , 1 , 1 , H 8 , 8 , 3 , 5 , 1 , 1 , H 8 , 8 , 5 , 3 , 1 , 1 , H 10 , 2 , 3 , 9 , 1 , 1 , H 10 , 2 , 5 , 7 , 1 , 1 , H 10 , 2 , 7 , 5 , 1 , 1 , H 5 , 3 , 3 , 3 , 3 , 3 , 3 , 3 , H 6 , 2 , 3 , 3 , 5 , 5 , 1 , 1 , H 6 , 2 , 3 , 5 , 3 , 5 , 1 , 1 , H 6 , 2 , 5 , 3 , 3 , 5 , 1 , 1 , H 6 , 4 , 3 , 3 , 3 , 5 , 1 , 1 (B.23) 67 The above bases are complete. For the foll owing b asis we mis s the two elements at depth 9 due to limited compu ter r esources. Y et the const ruction based on L 27 allows us to predict the last two elements: P 27 = H 27 , H 11 , 7 , 9 , H 13 , 11 , 3 , H 15 , 3 , 9 , H 15 , 5 , 7 , H 15 , 7 , 5 , H 15 , 9 , 3 , H 17 , 5 , 5 , H 17 , 7 , 3 , H 19 , 3 , 5 , H 19 , 5 , 3 , H 21 , 3 , 3 , H 7 , 5 , 5 , 7 , 3 , H 7 , 5 , 7 , 3 , 5 , H 7 , 7 , 3 , 7 , 3 , H 7 , 7 , 7 , 3 , 3 , H 9 , 3 , 9 , 3 , 3 , H 9 , 5 , 3 , 5 , 5 , H 9 , 5 , 3 , 7 , 3 , H 9 , 5 , 5 , 3 , 5 , H 9 , 5 , 5 , 5 , 3 , H 9 , 5 , 7 , 3 , 3 , H 9 , 7 , 3 , 3 , 5 , H 9 , 7 , 3 , 5 , 3 , H 9 , 7 , 5 , 3 , 3 , H 9 , 9 , 3 , 3 , 3 , H 11 , 3 , 3 , 3 , 7 , H 11 , 3 , 3 , 5 , 5 , H 11 , 3 , 3 , 7 , 3 , H 11 , 3 , 5 , 3 , 5 , H 11 , 3 , 5 , 5 , 3 , H 11 , 3 , 7 , 3 , 3 , H 11 , 5 , 3 , 3 , 5 , H 11 , 5 , 3 , 5 , 3 , H 11 , 5 , 5 , 3 , 3 , H 11 , 7 , 3 , 3 , 3 , H 13 , 3 , 3 , 3 , 5 , H 13 , 3 , 3 , 5 , 3 , H 13 , 3 , 5 , 3 , 3 , H 13 , 5 , 3 , 3 , 3 , H 15 , 3 , 3 , 3 , 3 , H 10 , 8 , 7 , 1 , 1 , H 10 , 10 , 5 , 1 , 1 , H 12 , 2 , 11 , 1 , 1 , H 12 , 4 , 9 , 1 , 1 , H 12 , 6 , 7 , 1 , 1 , H 12 , 8 , 5 , 1 , 1 , H 16 , 2 , 7 , 1 , 1 , H 5 , 3 , 5 , 3 , 5 , 3 , 3 , H 5 , 5 , 3 , 3 , 3 , 5 , 3 , H 5 , 5 , 3 , 3 , 5 , 3 , 3 , H 5 , 5 , 3 , 5 , 3 , 3 , 3 , H 5 , 5 , 5 , 3 , 3 , 3 , 3 , H 7 , 3 , 3 , 3 , 3 , 3 , 5 , H 7 , 3 , 3 , 3 , 3 , 5 , 3 , H 7 , 3 , 3 , 3 , 5 , 3 , 3 , H 7 , 3 , 3 , 5 , 3 , 3 , 3 , H 7 , 3 , 5 , 3 , 3 , 3 , 3 , H 9 , 3 , 3 , 3 , 3 , 3 , 3 , H 6 , 4 , 5 , 5 , 5 , 1 , 1 , H 6 , 6 , 3 , 5 , 5 , 1 , 1 , H 6 , 6 , 5 , 3 , 5 , 1 , 1 , H 6 , 6 , 5 , 5 , 3 , 1 , 1 , H 8 , 2 , 3 , 5 , 7 , 1 , 1 , H 8 , 2 , 3 , 7 , 5 , 1 , 1 , H 8 , 2 , 5 , 3 , 7 , 1 , 1 , H 8 , 2 , 5 , 5 , 5 , 1 , 1 , H 8 , 2 , 5 , 7 , 3 , 1 , 1 , H 8 , 2 , 7 , 3 , 5 , 1 , 1 , H 8 , 2 , 7 , 5 , 3 , 1 , 1 , H 8 , 4 , 3 , 3 , 7 , 1 , 1 , H 7 , 5 , 7 , 5 , 3 → ? H 6 , 4 , 6 , 4 , 3 , 1 , 1 , 1 , 1 , H 7 , 5 , 3 , 3 , 3 , 3 , 3 → ? H 6 , 4 , 3 , 3 , 3 , 3 , 3 , 1 , 1 W e ha ve selected the last two elements for the necessary extension on t he basis of the Appendix in the thesis by Racinet [68] in which for th ese two elements the numbers 6 and 4 seem to play a special role. Although we have also results for P 28 in whi ch t he l eading depth is mis sing, there are too many elements missing to give a reliable list of the basis elements . It should be remarked though that also for P 28 we expect a 2-fold pushdo wn fr om depth 8 to depth 4. C Explicit pushdowns Belo w we list all push downs up to w = 21 and one at w = 2 2 with the mixi ng with terms of equal weight and depth in the left h and side and all remain ing Euler sums in the right hand side. The functi on A is deﬁned in (10.3). W e only list that part o f t he pushdowns that we consider particularly in teresting. The complete formulas can b e found i n th e data mine in the programs part. The name of the ﬁle is pushdowns. h . Z 6 , 4 , 1 , 1 = − 64 27 A 7 , 5 + · · · (C.1) Z 6 , 4 , 3 , 1 , 1 = 1408 81 A 7 , 5 , 3 + · · · (C.2) Z 8 , 6 , 1 , 1 + 542 175 Z 5 , 5 , 3 , 3 − 19 7 Z 7 , 3 , 3 , 3 = − 1024 405 A 9 , 7 + · · · (C.3) 68 Z 6 , 6 , 3 , 1 , 1 − 14 5 Z 5 , 3 , 3 , 3 , 3 = 5120 243 A 7 , 7 , 3 + · · · (C.4) Z 10 , 6 , 1 , 1 − 10 3 Z 9 , 3 , 3 , 3 − 124 35 Z 7 , 3 , 5 , 3 − 124 35 Z 7 , 3 , 3 , 5 − 3282 875 Z 5 , 5 , 5 , 3 = − 8192 3375 A 11 , 7 + · · · (C.5) Z 6 , 4 , 3 , 3 , 1 , 1 = − 392 27 A 7 , 5 , 3 , 3 + · · · (C.6) Z 8 , 6 , 3 , 1 , 1 − 61 7 Z 7 , 3 , 3 , 3 , 3 + 1774 175 Z 5 , 5 , 3 , 3 , 3 + 2 5 Z 5 , 3 , 5 , 3 , 3 = 647168 34263 A 7 , 7 , 5 + 45056 1215 A 9 , 7 , 3 + · · · (C.7) Z 6 , 6 , 5 , 1 , 1 + 13 Z 7 , 3 , 3 , 3 , 3 − 268 25 Z 5 , 5 , 3 , 3 , 3 + 6 5 Z 5 , 3 , 5 , 3 , 3 = − 3598336 125631 A 7 , 7 , 5 − 759808 4455 A 9 , 7 , 3 + · · · (C.8) Z 10 , 8 , 1 , 1 − 13 2 Z 11 , 3 , 3 , 3 − 304 45 Z 9 , 3 , 3 , 5 − 3601 525 Z 9 , 3 , 5 , 3 − 3799 525 Z 7 , 3 , 5 , 5 + 1371 196 Z 7 , 7 , 3 , 3 + 163 2450 Z 7 , 5 , 5 , 3 = − 16384 6615 A 11 , 9 + · · · (C.9) Z 6 , 4 , 3 , 5 , 1 , 1 − 68 5 Z 5 , 3 , 3 , 3 , 3 , 3 = − 118784 243 A 9 , 5 , 3 , 3 − 2560 243 A 7 , 5 , 3 , 5 + · · · (C.10) Z 8 , 4 , 3 , 3 , 1 , 1 − 28 5 Z 5 , 3 , 3 , 3 , 3 , 3 = 32768 81 A 9 , 5 , 3 , 3 − 10240 2187 A 7 , 5 , 3 , 5 + · · · (C.11) Z 8 , 6 , 5 , 1 , 1 − 68 9 Z 9 , 3 , 3 , 3 , 3 − 832 105 Z 7 , 3 , 5 , 3 , 3 − 967 105 Z 7 , 3 , 3 , 5 , 3 − 1042 105 Z 7 , 3 , 3 , 3 , 5 − 13182 875 Z 5 , 5 , 5 , 3 , 3 − 6 7 Z 5 , 5 , 3 , 5 , 3 = − 194240512 9628875 A 9 , 7 , 5 − 229376 1125 A 11 , 7 , 3 − 80972546048 337010625 A 11 , 5 , 5 + · · · (C.12) Z 10 , 4 , 5 , 1 , 1 − 46 9 Z 9 , 3 , 3 , 3 , 3 − 67 21 Z 7 , 3 , 5 , 3 , 3 − 73 21 Z 7 , 3 , 3 , 5 , 3 − 79 21 Z 7 , 3 , 3 , 3 , 5 − 482 175 Z 5 , 5 , 5 , 3 , 3 − 46 175 Z 5 , 5 , 3 , 5 , 3 = + 15966208 641925 A 9 , 7 , 5 + 32768 2025 A 11 , 7 , 3 − 1691951104 67402125 A 11 , 5 , 5 + · · · (C.13) 69 Z 10 , 6 , 3 , 1 , 1 − 46 9 Z 9 , 3 , 3 , 3 , 3 − 632 105 Z 7 , 3 , 5 , 3 , 3 − 86 15 Z 7 , 3 , 3 , 5 , 3 − 572 105 Z 7 , 3 , 3 , 3 , 5 − 4792 875 Z 5 , 5 , 5 , 3 , 3 + 46 175 Z 5 , 5 , 3 , 5 , 3 = + 124608512 9628875 A 9 , 7 , 5 + 16384 10125 A 11 , 7 , 3 − 758235136 48144375 A 11 , 5 , 5 + · · · (C.14) Z 6 , 4 , 3 , 3 , 3 , 1 , 1 = − 5120 81 A 7 , 5 , 3 , 3 , 3 + · · · (C.15) Z 12 , 8 , 1 , 1 + 13598459235 18816311591 Z 7 , 5 , 7 , 3 − 9790486696 6109192075 Z 9 , 3 , 5 , 5 − 3021879830 2688044513 Z 9 , 3 , 7 , 3 − 560739181022 201603338475 Z 9 , 5 , 3 , 5 − 19968330538 13440222565 Z 9 , 5 , 5 , 3 − 66543918797 40320667695 Z 9 , 7 , 3 , 3 + 2598592817 707380135 Z 11 , 3 , 3 , 5 − 3186058443 2688044513 Z 11 , 3 , 5 , 3 − 20352278271 13440222565 Z 11 , 5 , 3 , 3 + 7925677546 1221838415 Z 13 , 3 , 3 , 3 = − 524288 212625 A 13 , 9 + · · · (C.16) The + · · · i ndicates t erms that are purely MZVs o f lower depth or products of lo wer weight MZVs. The complet e relations can hav e up to about 150 terms. Hence we giv e them in a ﬁle in th e data mine. T he ﬁrst 15 of these relations were derive d with the help of PSLQ and/or the LLL algorithm. Seven of them could be deriv ed with the data m ine. Unfortunately for depth d = 5 objects we hav e only exac t re sults up to we ight w = 17 and for depth d = 4 we have only e xact results up to weight w = 22 . The abov e results used th e ava ilable resou rces t o th eir lim it. The formula in (C.15) needed 45 hours of runni ng time using the LLL algorithms as implem ented i n PARI in a 152 parameter search at 8000 digits and was check ed afterwards at 10000 digits. W e ha ve e xpressed the pu shdowns in terms of the A -function that has the same in- dices as the element of L w that was extended. It is not clear whether this scheme can be maintained for pushdowns beyond the ones we present. Some A -functions cannot be used because they express directly in terms of equal or lower depth MZVs. This then has again inﬂuence on the s election o f the basis. In the end it may be that we have to drop one or more requi rements for the basis. A simple example of such an A -function exists already at weight w = 15 : A 7 , 3 , 5 = + 7649 143360 Z 7 , 3 , 5 − 7089 143360 ζ 5 Z 7 , 3 − 2097 71680 ζ 3 5 − 3429 5120 ζ 7 Z 5 , 3 − 116396017 2867200 ζ 15 + 1083797 40960 ζ 2 ζ 13 + 81059 71680 ζ 2 2 ζ 11 − 110993 627200 ζ 3 2 ζ 9 − 43311 448000 ζ 4 2 ζ 7 − 27831 78400 ζ 5 2 ζ 5 . (C.17) 70 It is also possible t o express each pushdown in terms of a single Euler sum rather than an A -functi on. In a sense th is is less tel ling. After all the A -function contains half of th e terms of the doubling relation and the doubl ing relations seem to be at t he origin of the pushdowns. Also we cou ld not ﬁnd much structure concerning which Euler sum(s) to select. There are often m any possibiliti es. In the case of the A -functions on e can make a unique selection : t he A -functio n should have th e same index ﬁeld as the elem ent of the set L w that represents t he p ushdown. Anywa y , for completeness we giv e here a sing le E uler sum for eac h of the pus hdowns. W e ha ve dropped all f actors and terms which ha ve MZVs of the same weight or products of MZVs with lower weight. H -representation Z -representation A 7 , 5 → H − 9 , 3 Z − 9 , − 3 A 7 , 5 , 3 → H − 6 , − 3 , 6 Z − 6 , 3 , − 6 A 9 , 7 → H − 13 , 3 Z − 13 , − 3 A 7 , 7 , 3 → H − 6 , − 5 , 6 Z − 6 , 5 , − 6 A 11 , 7 → H − 15 , 3 Z − 15 , − 3 A 7 , 5 , 3 , 3 → H 6 , − 5 , 4 , 3 Z 6 , − 5 , − 4 , 3 A 9 , 7 , 3 → H − 8 , − 3 , 8 , H − 6 , − 7 , 6 Z − 8 , 3 , − 8 , Z − 6 , 7 , − 6 A 7 , 7 , 5 → H − 8 , − 3 , 8 , H − 6 , − 7 , 6 Z − 8 , 3 , − 8 , Z − 6 , 7 , − 6 A 11 , 9 → H − 17 , 3 Z − 17 , − 3 A 7 , 5 , 3 , 5 → H 8 , − 5 , 4 , 3 , H 6 , − 5 , 6 , 3 Z 8 , − 5 , − 4 , 3 , Z 6 , − 5 , − 6 , 3 A 9 , 5 , 3 , 3 → H 8 , − 5 , 4 , 3 , H 6 , − 5 , 6 , 3 Z 8 , − 5 , − 4 , 3 , Z 6 , − 5 , − 6 , 3 A 9 , 7 , 5 → H − 8 , − 5 , 8 , H − 6 , − 9 , 6 , H − 8 , − 3 , 10 Z − 8 , 5 , − 8 , Z − 6 , 9 , − 6 , Z − 8 , 3 , − 10 A 11 , 5 , 5 → H − 8 , − 5 , 8 , H − 6 , − 9 , 6 , H − 8 , − 3 , 10 Z − 8 , 5 , − 8 , Z − 6 , 9 , − 6 , Z − 8 , 3 , − 10 A 11 , 7 , 3 → H − 8 , − 5 , 8 , H − 6 , − 9 , 6 , H − 8 , − 3 , 10 Z − 8 , 5 , − 8 , Z − 6 , 9 , − 6 , Z − 8 , 3 , − 10 A 7 , 5 , 3 , 3 , 3 → H 3 , − 6 , − 3 , 6 , 3 Z 3 , − 6 , 3 , − 6 , 3 A 13 , 9 → H − 19 , 3 Z − 19 , − 3 Of course more complete results can be found in the data mine. 71 References [1] L. Euler , Medit a tiones cir ca singular e serium genus , Novi Comm. Acad. Sci. Petropol. 20 (1775) 140–186, reprinted in Opera Omnia ser I vol. 15, (B.G. T eubner , Berlin, 1927), 217–267. [2] D. Zagier , V a l ues of zeta functions and their applicatio ns , in : First Eu ropean Congress of Mathematics, V ol. II, (Par is, 1992 ), Progr . M ath., 120 , (Birkh ¨ auser , Basel–Boston, 1994), pp. 497–512. [3] For an e x tended list of refe rences, see : M.E. Hof fm an’ s page http://www .usna.edu/ Users/math / ∼ meh/biblio.html . [4] P . H. Fuss (ed.), Corr espondance Math ´ ematique et Physique de quelq ues c ´ el ` ebr es G ´ eom ` etr es (T ome 1), St. Petersbur g, 1843 ; N. Ni elsen, Die Gammaf unktion (Chelsea, New Y ork, 1965), Reprint of Handbuch der Theorie der Gammafu n kt i on (T eubn er , Leipzig, 1906). [5] S. Fischler , Irrationalit ´ e de valeurs de z ´ eta , S ´ em. Bourbaki, Novembre 2002, exp. n o. 91 0 , Asterisque 294 (2004) 27–62, http://www .math.u-ps ud.fr/ ∼ fischler/publi.html ; P . Colmez, Arithmetique de la fonction z ˆ eta , in: Journ ´ ees X-UPS 2002. La fontion z ˆ eta. Edi tions de l’Ecole polytechniqu e, Paris, 2002, 37–164, http://www .math.poly technique. fr/xups/vol02.html ; M. W alds chmidt, Multiple P olylogarit h m s : an Intr oduction , Number Th eory and Discrete Math ematics, Edit ors: A.K. Agarwal, B .C. Berndt, C.F . Kratt enthaler , G.L. Mullen, K. Ramachandra and M. W ald schmidt, (Hin dustan Book Agency , 2002 ), 1–12; M. W aldschm idt, V al eurs z ˆ eta multiples. Une intr oducti o n , Journal de th ´ eorie des nombres de Bordeaux, 12 (2) (2000 ) 581–595; M. Hut tner and M. Petitot, Arith me ´ etique des fonctio ns d’zetas et Ass o ciateur de Drinfel’d , (UFR d e Math ´ emati ques, Lille, 2005); C. Hertling, A G Mannheim -Heidelberg, SS2007. [6] P . Cartier , F onction s po l ylogarithmes, nombr es pol yz ˆ etas et groupes pr o-unipotents , S ´ em. Bourbaki, Mars 2001, 53 e ann ´ ee, exp. no. 885 , As terisque 282 (2002 ) 137– 173. [7] V .V . Z udilin, A l gebraic r elations for mult iple zeta values , Uspekhi Mat. Nauk 58 (1) 3–22. [8] R. Barbieri, J. A. Mign aco and E. Remiddi, Electr o n form-factors up to f ourth order . 1 , Nuov o Cim. A 11 (1972) 824–864; M. J. Le vine, E. Remiddi and R. Roskies, Analytic Contributions T o The G F actor Of The Electr on In Sixth Or d er , Phys. Re v . D 20 (197 9) 2068–2076; A. Dev oto and D. W . Duke, T able Of Inte grals And F ormulae F or F e ynman Diagram Calculations , Ri v . Nuov o C im. 7N6 (198 4) 1–39. 72 [9] A. Gonzalez-Arroyo, C. Lopez and F . J. Yndu rain, Second Or der Contributions T o The Structu re Functions In Deep Inelastic S cattering. 1. Theor etical Calculations , Nucl. Phys. B 153 (1979) 161–18 6. [10] J. A. M. V ermaseren, Harmonic sums , Melli n transforms and inte grals , Int. J. Mod. Phys. A 14 (1999) 2037–20 76, [arXiv:hep -ph/980628 0 ]. [11] J. Bl ¨ umlein and S. Kurth, Harmonic su ms and Mellin transforms up to two-loop or der , Phys. Re v . D 60 (1999) 014018, 31 p., [arXiv:hep -ph/981024 1 ]. [12] D. J. Broadhurst, On t he enumeration of irreducible k-fold Euler sums and their r oles i n knot th eor y and ﬁeld theory , arXiv:hep- th/9604128 . [13] D. J. Broadhurst and D. Kreimer , Associati on of multiple zeta values with posi- tive knot s via F eynman diagrams up to 9 loops , Phys. Lett. B 393 (1997) 403–412, [arXiv:hep -th/960912 8 ] . [14] A.B. Goncharov , Multiple polylogarithms and mixed T ate moti ves , arxiv:math .AG/010305 9 ; T . T erasoma, Mi xed T ate Motives and Multiple Zeta V alues , In vent. M ath. 1 49 (2) (2002) 339–369, arxiv:math .AG/010423 ; P . Deligne and A.B. G oncharov , Gr oupes fondament aux motiviques de T ate mixtes , Ann. Sci. Ecole Norm. Sup., S ´ erie IV 38 (1) (20 05) 1–56. [15] H.R.P . Ferguson and D.H. Bailey , D. H. A P ol ynomial T ime, Numeri cal ly Stab l e Inte ger Relat ion Algorithm , RNR T echn. Rept. RNR-91-032, Jul. 14, 199; D. H. Bailey and D.J. Broadhurst, P arallel i nte ger re lation det ecti o n: techniques a nd application s , Math. Comp. 70 (2001), no. 236, 1719–1736 (electronic), [arxiv: math.NA/99 05048 ] . [16] A. K. Lenstra, H. W . Lenstra, and L. Lov asz, F actori ng P olynomials with Rati o nal Coef ﬁcients , Math. Ann. 261 (1982) 5 15-534. [17] H. Poincar ´ e, Sur l es gr ou pes d‘ ´ equations l in ´ eair es , Acta Math. 4 (1984) 201– 311; J.A. L appo-Danielevsk y , M ´ emoirs sur la Th ´ eorie des Syst ` emes Diff ´ er entiels Lin ´ eair es , (Chelsea, New Y ork, 1953); K.T . Chen, F ormal differ enti al equations , Ann. of Math . (2) 73 (1961 ) 110–133; Iterated inte g rals of di ffer enti a l for m s and loop space hom o logy , Ann. Math. 97 (1973) 217–246. [18] E. Remiddi and J . A. M . V ermaseren, Harmonic polylogari t hms , Int. J. M od. Phys. A 15 (2000) 725– 754, [arXiv:hep -ph/990523 7 ]. [19] J.M. Borwein, D.M. Bradley , and D.J . Broadhurst, Evaluation of k-fold Eu l er/Zagier sums: a compendium of r esults f o r arbitrary k , Elec. J. Combin. 4 (199 7) no.2 #R5, hep-th/961 1004 . [20] http://www .nikhef.nl / ∼ form/datam ine/datamine.html 73 [21] J. A. M. V er maseren, New featur es of FORM , arXiv:math -ph/001002 5 . [22] M. T entyu kov and J. A. M. V ermaseren, The mu ltithr eaded version of FORM , arXiv:hep- ph/0702279 . [23] S. Moch, J. A. M. V er maseren and A. V ogt, The thr ee-loop splittin g func- tions in QCD: The non-singlet cas e , Nucl. Phys. B 68 8 (2004) 101–134, [arXiv:hep -ph/040319 2 ] ; A. V ogt, S. Moch and J. A. M. V ermaseren, The t h r ee-loop splitt ing fun c- tions in QCD: The singlet case , Nucl. Phys. B 691 (200 4) 129 –181, [arXiv:hep -ph/040411 1 ] ; J. A. M . V ermaseren, A. V ogt and S. Moch, The thi r d-or der Q CD corr ecti o ns to deep-inelastic scattering by photo n e xchange , Nucl. Phys. B 724 (2005) 3 –182, [arXiv:hep -ph/050424 2 ] . [24] J. Bl ¨ umlein, A. De Freitas, W . L. van Neerven and S. Klein, The lon g itudinal heavy quark structu r e function F Q Q L in the r e gion Q 2 ≫ m 2 at O ( α 3 s ) , Nucl . Phys. B 755 (2006) 272 [arXiv:hep-ph/0608024 ]; I. Bierenbaum, J . Bl ¨ uml ein and S. Klein , Mellin Mom ent s of the O ( α 3 s ) Heavy Flavor Contributions to unpolarized Deep-Inelastic Scattering at Q 2 ≫ m 2 and Anomalous Dimensions , arXi v:0904.3563 [hep-ph], Nucl. Phys. B820 (2009) 417; J. Bl ¨ um lein, M. Kauers, S. Klein and C. Schneider , Det erm i ning the closed forms of the O ( a 3 s ) anomalous dimensio n s and W ilson coefﬁcients fr om Mellin moment s by means of computer algebra , arXi v: 0902.4091 [hep-ph]; Comp. Phys. Commun. (2009) in print; [25] J. Bl ¨ umlein, Str uctural Relations of Harmonic Sums and Mell in T ransforms up t o W eight w = 5 , arXiv:0901.3106 [he p-ph]; Comp. Phys. Commun. (2009) in print; J. Bl ¨ uml ein and S. Klein, St r uctural Relation s between Harmoni c Sums up to w=6 , arXiv:0706 .2426 [hep-ph] , 5 p. [26] J. Bl ¨ umlei n, Structural Relations of Harmonic S u ms and Mellin T ransforms at W eight w=6 , arXiv:0901.0837 [ math-ph], Clay Mathematical Institute Proceedings, in print. [27] D. J. Broadhurst, J. A. Gracey and D. Kreimer , Be yon d the t r iangle and unique- ness relations: Non-zeta counterterms at la rg e N fr om p ositive knots , Z. Phys. C 75 (1997) 559–574, [arXi v:hep-th/96071 74 ]. [28] D. J. Broadhurst, Massive 3-lo op F ey nman diagrams r educib le to SC* pri mi- tives of algebras o f the s i xth r oot of unity , Eur . Phys . J. C 8 (1999) 311 –333, [arXi v:hep-th/98030 91 ]. [29] Y . Andre, Ambiguity Theory , Old and New , t o appear i n: Bollettin o U.M.I. (8) I (2008) and talk at the Motives, Quantum Fiel d Theor y , and Pse udodiff erential Oper- ators Boston Univ ersi ty , June 2–13, 2008. 74 [30] F . Brown, The massless higher-loop two-point functio n , Commun. Mat h. Phys. 287 (2009) 925-958, [arXiv:080 4.1660 [math.AG]] . [31] M. Kontse vich and D. Zagier , P eri o ds , Mathematics Unlimited-2001 and Beyond, (Springer , Berlin, 2001), 771–808. [32] C. Bogner and S. W einzierl, J. Math. Phys. 50 (2009) 04 2302 [hep-th]]. [33] L. Lewin, P olylogarith ms and Associated Functions , (North Holland, Ne w Y ork, 1981). [34] N. Nielsen, Der Eulersche Dilogarithmu s und seine V erallgemeinerungen , Nova Acta Leopoldina, XC (3)(1909) 125–2 11; K. S. K ¨ ol big, Nielsen’ s Generalized P olylogarithms , SIAM J . Math. Anal. 1 7 (1986) 1232–1258. [35] S. Moch, P . Uwer and S. W einzierl, Nested sums, expansion of transcendental func- tions and mult i -scale multi-loop inte grals , J. Math. Phys. 43 (2002) 336 3–3386, [arXi v:hep-ph/011008 3 ]. [36] A.B. Goncharov , Multiple polylogarith m s , cyclotomy and modu l ar complexes , Math. Res. Lett. 5 (1998) 497– 516. [37] J. Bl ¨ umlein, Algebraic r ela tions between ha rmonic sums and associat ed quantities , Comput. Phys. Commun. 159 (2004) 19–54, [arXiv:hep -ph/031104 6 ]. [38] R.C. L yndon, On Burnsides pr ob l em , Tra ns. Am er . Math. Soc. 77 (1954) 2 02–215; On Bur nsides pr obl em II , T rans. Amer . Math. Soc. 78 (1955) 329–332 ; C. Reutenauer , F r ee Algebras , (C alendron Press, Oxford, 1993). [39] M.E. H of fman, Algebraic Aspects of Multipl e Zeta V alues , in: Zeta Functi ons, T opol- ogy and Quantum Ph ysics, De velopments in Mathematics, 14 , T . Aoki et. al. (eds.), Springer , New Y ork, 2 005, pp. 51–74, arXiv:math /0309452 [math.QA] . [40] M.E. Hof fman, Multiple harmo n i c series , Paciﬁc J. Math. 152 (1992) 275–290. [41] A. Hurwitz, Einige Eigenschaften der Dirichlet’ s chen Funktionen F ( s ) = ∑ ( D / n ) · 1 / ( n s ) , di e bei der Best immung der Klassenanzahlen Bin ¨ ar er qu a dratischer F ormen auftr eten , Z. f ¨ ur Math. und Physik 27 (1882) 8 6–101; H. Hasse, Ein Summierungs verf a hr en f ¨ ur die Riemannsche ζ –Reihe , Math. Z. 32 (1930) 458–464. [42] Leonardus Pisanus de ﬁliis Bonaccij, Liber aba ci , Cap. 12.7, (Pisa, 1202); L.E. Sigler , F ibon acci’ s Liber Ab aci , (Springer , Berlin, 2002). [43] see http://en. wikipedia. org/wiki/P adovan sequence ; http://www .emis.de/j ournals/NN J/conferences/N2002-Padovan.html 75 [44] E. W itt, T r eue Darstellung Liesc her Ringe , Jo urn. Reine Angew . Mathematik 177 (1937) 152–160. Die Unterring der fr eien Lieschen Ringe , Math. Zeitschr . 64 (1956) 195–216. [45] R. Perrin, Item 1484 , L ’Interm ´ edi are des Math. 6 , (1899) 76–77; A. W illi ams and D. Shanks, Str ong primality t ests that ar e not sufﬁcient , M athemat- ics of Computation 39 (159)(1982) 255–300 . [46] E. Lucas, Th ´ eorie des fo nctions nu m ´ eriques simplement p ´ eriodiques , American Journal of Mathematics 1 (1878) 197– 240. [47] J. M. Borwein, D. M. Bradley , D. J. Broadhurst and P . Lisonek, Special V al- ues of Mul t iple P olylogarit hms , T rans. Am. Mat h. Soc. 353 (200 1) 907–941, [arXiv:mat h/9910045 ] [48] J.M. Borwein and R. Girgensohn, Eval uation of T ripl e Euler Sums , The electronic journal of combinatorics 3 (1996), #R23. [49] C. M arkett, T ripl e sums and the Riemann zeta fun ction . J. Number Theory 4 8 (1994), 113–132. [50] M. Espie, J.-C. Novelli, and G. Racinet, F ormal Computations about Multi p l e Zeta V alues , Preprint SFB-478 Univ . M ¨ unster Heft 260 (2003), in : From Com binatoric s to Dynamical Systems , Strasbourg, 2002, IMRA Lect. Math. Theor . Phys. 3 , F . Fauve t and C. Mitschi (eds.), (de Gryter , Berlin, 2003), 1–16. [51] M. Kaneko, M. Noro and K. Tsurumaki, On a conjecture for the dimension of the space of the multiple zeta values , Software for Alg ebraic Geom etry , IMA 148 (2008), 47–58. [52] M.E. Hoffmann and Y . Ohno, Relat ions of mul tiple zeta values and their algebraic e xpr essions , J. Algebra 262 (2003) 332–347. [53] Sun Tzu, Sun Zi suanjing , 3rd century AD (between 280 and 473 AD); re-published by Qin Jiushao in: Shu s hu Jiuzhang , 1247. [54] R. Gastm ans and W . Troost, On the evaluatio n of polylogarithmic inte grals , Sim on Ste vin, 55 (1981), 205–21 9, KUL-TF-80/10, MR 0647134 (83c:65028). [55] D.J. Broadhurst, Data Base to w = 9 , pri va tely distributed; D. Zagier , priv ate communicati on. [56] The P ARI/GP page: http://par i.math.u-b ordeaux.fr / [57] D.H. Bailey , J.M. Borwein, and R. Girgensohn, Experimental Evaluat ion of E u ler Sums , Experiment. Math. 3 (1994), 17–30 . [58] M. Bigotte, G. Jacob, N.E. Oussous, and M. Petitot, T a b l es des r elations de la fonc- tion z ´ eta color ´ ee avec 1 racine , LIFL USTL Lille preprint IT –332, Nov . (19 98). Non-alternating part: pp. 1–22 7; alternating part: pp. 1–274 . O nly the ﬁnite va lues are tab ulated. 76 [59] M. Bigot te, G. Jacob, N.E. Oussous, and M. Petit ot, L yndon wor ds and shufﬂe algen- ras for generating the colo u r ed multiple zeta valu es r elati on tables , Theor . Comp. Sci. 273 (2002) 271– 282. [60] H.N. Minh and M. Petitot, Lyndon wor ds, polylogarithms and t he Rieman n ζ func- tion , Discrete Maths. 217 (2000) 273–292 . [61] El W ardi - m ´ em oire DEA, Lille 1, Juillet, 1999. [62] H.N. Minh, G. Jacob, N.E. Oussuou s and M. Petitot, Aspects combi natoir es des polylogarithmes et des sommes d’Euler- Zagier , J . ´ Electr . S ´ em. Lothar . Combin. 43 (2000), Art. B43e, 29 pp. [63] J.A.M. V e rmaseren, http://www .nikhef.nl / ∼ t68/FORMap plication , Nov . 2003. [64] H.N. Minh, pri vate comm unication, June 2008. [65] J.A.M. V e rmaseren, http://www .nikhef.nl / ∼ form . [66] J.A.M. V ermasere n, T uning form wit h large calculat ions. Nucl. Phys. Proc. Suppl. 116: 343-347 ,2003. [67] J.A.M.V ermaseren and J.V ollinga, in prepartion. [68] G. Racinet, S ´ eries g ´ en ´ eratrices non-commutatives de polyz ˆ etas et associat eurs de Drinfel’d , Ph.D. T hesis, Amiens, France, 2000. [69] M.E. Hoffman, The Algebra of Multiple Harmonic Series , Journ. of Algebra 194 (1997) 477–495. [70] D. B owman and D. M . B radley , T he algebra and combinat orics of shufﬂes and mul- tiple zeta values , J. Combin. Theory Ser . A97 (2002) 43=-61. [71] J. Zhao, Double Shufﬂe Relations of Euler Sums , arXiv:0705 .2267 [math.NT] . [72] M. E. Hoffman and C. Moen, S u ms of tripl e harmonic series , J. Number Theory 60 (1996), 329–331; A. Gran vil le, A decomposi t ion of Riemann’ s zeta-f u nction , in Analyt ic Number The- ory , London M athematical Society Lecture Note Series 247, Y . Motohashi (ed.), (Cambridge Unive rsity Press, Cambridge, 1997) pp. 95–101; D. Zagier , Multipl e zeta v alues, preprint. [73] J. Ok uda and K. Ueno, The Sum F ormula of Multiple Zeta V alues and Connc- tion Pr oblem of the F ormal Knizhn i k–Zamolodchikov Equat ion , in: Ze ta Functio ns, T opology an d Quantum Physics, De velopments i n Mathematics, 14 , T . Aoki et. al. (eds.), Springer , New Y ork, 2005, pp. 145–170, arXiv:math /0310259 [math.NT] . [74] K. Ihara and M. Kaneko, A note on r elati o ns of mu l tiple zeta values , p reprint. 77 [75] T .Q .T . Le and J. Murakami, K ont s evic h’ s i nte gral for the Homﬂy polynomial and r elation s between valu es of mul tiple zeta functions , T opology Appl. 62 (1995 ) 193– 206. [76] Y . Ohno, A generalization of t he duality and sum formulas on the mul tiple zeta values , J. Number Theory , 74 (19 99) 189–209. [77] Y . Ohno and D. Zagier , Mu ltiple zeta values of ﬁxed weight, depth, and height , Indag. Math. (N.S.) 12 (2001 ) 483–487. [78] Y . Ohno and N. W akabayashi, Cyclic s u m of mult iple zeta values , ’ Acta Arithmetica 123 (2006), 289 –295. [79] K. Ihara, M. Kaneko, and D.Zagier , Derivat ion and doubl e shufﬂe re lations for mul- tiple zeta values , Compositio Math. 142 (2006) 307– 338; preprint MPIM2004-100. [80] J. ´ Ecalle, Th ´ eorie des moul es . 3 vol, pr ´ epubl ications math ´ ematiques d‘Orsay , 1981, 1982, 1 985; La libr e g ´ en ´ eration des mu l tic ˆ etas et leur d’ecomposition canonico- e xplicite en irr ´ eductibles , (automne 1999); Ari /gari et la d ´ ecomposition des mul- tiz ˆ etas en irr ´ eductibles , Pr ´ epublication, avril 2000. [81] J. Zhao, Linear Relations o f Special V alues of Multiple P olylogarith m s at Roots of Unity , arXiv:0707 .1459 [math.NT] . [82] H. Tsunogai, On ranks of the s table derivation algebra and Delinge’ s pr o blem , Proc. Japan Acad. Ser . A M ath. Sci. 73 (1997) 29–31; H. Furusho, The Multipl e Zeta V alue Algebra and the Stable Derivat ion Algebra , Publ. RIMS, Kyoto Univ . 39 (2003) 695–720. [83] G.H. Hardy and E.M. Wright, An Intr odu ction to the Theory of Numbers , (Calendron Press, Oxford, 2002). 78

The Multiple Zeta Value Data Mine

Original Paper

Comments & Academic Discussion

Leave a Comment

Original Paper

Related Papers

Comments & Academic Discussion

Leave a Comment