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APPEARED IN BULLETIN OF THE

arXiv:math/9210214v1 [math.NT] 1 Oct 1992APPEARED IN BULLETIN OF THEAMERICAN MATHEMATICAL SOCIETYVolume 27, Number 2, October 1992, Pages 273-278ADDITIVE FUNCTIONS ON SHIFTED PRIMESP. D. T. A. ElliottAbstract.

Best possible bounds are obtained for the concentration function of anadditive arithmetic function on sequences of shifted primes.A real-valued function f defined on the positive integers is additive if it satisfiesf(rs) = f(r) + f(s) whenever r and s are coprime. Such functions are determinedby their values on the prime-powers.For additive arithmetic function f, let Ch denote the frequency amongst theintegers n not exceeding x of those for which h < f(n) ≤h + 1.

Estimates for Chthat are uniform in h, f, and x play a vital rˆole in the study of the value distribu-tion of additive functions. They can be employed to develop criteria necessary andsufficient that a suitably renormalised additive function possess a limiting distribu-tion, as well as to elucidate the resulting limit law.

They bear upon problems ofalgebraic nature, such as the product and quotient representation of rationals byrationals of a given type. In that context their quantitative aspect is important.It is convenient to write a ≪b uniformly in α if on the values of α beingconsidered the functions a, b satisfy |a(α)| ≤cb(α) for some absolute constant c.When the uniformity is clear, I do not declare it.LetW(x) = 4 + minλλ2 +Xp≤x1p min(1, |f(p) −λ log p|)2,where the sum is taken over the prime numbers.

Improving upon an earlier resultof Hal´asz, Ruzsa proved that Ch ≪W(x)−1/2, uniformly in h, f, and x ≥2 [9].This result is best possible in the sense that for each of a wide class of additivefunctions there is a value of h so that the inequality goes the other way.From a number theoretical point of view it is desirable to possess analogs ofRuzsa’s result in which the additive function f is confined to a particular sequenceof integers of arithmetic interest. In this announcement I consider shifted primes.Let a be a nonzero integer.

Let Qh denote the frequency amongst the primes pnot exceeding x of those for which h < f(p + a) ≤h + 1.1991 Mathematics Subject Classification. Primary 11K65, 11L20, 11N37, 11N60, 11N64.Received by the editors November 26, 1991.

Presented at the 1992 Illinois Number TheoryConference, University of Illinois at Urbana-Champaign, April 3–4, 1992c⃝1992 American Mathematical Society0273-0979/92 $1.00 + $.25 per page1

2P. D. T. A. ELLIOTTTheorem 1.

The estimate Qh ≪W(x)−1/2 holds uniformly in h, f, and x ≥2.If for an integer N ≥3 we define Sh to be the frequency amongst the primes pless than N of those for which h < f(N −p) ≤h + 1, and setY (N) = 4 + minλλ2 +Xp

IfE(x) = 4 +Xp≤xf(p)̸=01p,then Timofeev shows that the number of primes not exceeding x for which f(p +a) assumes any (particular) value is ≪π(x)E(x)−1/2(log E(x))2. Employing thepresent Theorem 1, the logarithmic factor may be stripped from this bound.

Theimproved inequality is then analogous to an estimate of Hal´asz concerning additivefunctions on the natural numbers and, in a sense, best possible [8].The concentration function estimate of Theorem 2 also has many applications,in particular, to the study of the value distribution of additive functions. These arenew and of a new type.

They involve not only the primes but also the length ofthe interval on which the additive function is considered. Thus the frequencies(π(N −1))−1Xp11p,X|f(p)|≤1f(p)p,X|f(p)|≤1f(p)2pconverge.

The latter is the classical condition of Erd¨os and Wintner required whenconsidering frequencies over the natural numbers [7]. More complicated examplesinvolving unbounded renormalisations of additive functions can also be successfullytreated.The method of this paper lends itself well to the study of the representation ofrationals by products and quotients of shifted primes.The proofs of Theorems 1 and 2 apply Fourier analysis.

Since the F´ejer kernelis nonnegative, Qh does not exceed(1)3π(x)−1 Xp≤xZ 1−1(1 −|t|)e−ithg(p + a) dt,

ADDITIVE FUNCTIONS ON SHIFTED PRIMES3where g(n) is the multiplicative function exp(itf(n)). To deal directly with themean value of g(p + a) over the primes would require finer information concerningthe distribution of primes in residue classes than is currently available.

Let 3|a| ≤w ≤z. Ultimately z will be chosen a power of x, w a power of log x.

Let P, R denotethe products of the primes in the ranges 3|a| < p ≤w, w < p ≤z respectively.I majorize (1) by introducing a Selberg square function (Pd|(n,R) λd)2, where theλd are real, zero if d > z, λ1 = 1.Expanding and interchanging the order ofsummation gives(2)Qh ≤3π(x)Z 1−1(1 −|t|)e−ith Xdj|Rλd1λd2Xn≤x,(n,P )=1n≡0 (mod [d1,d2])g(n + a) dt +3zπ(x).We are reduced to the study of multiplicative functions on arithmetic progressionswith moduli large compared to x.It may seem curious to retain the condition(n, P) = 1. However, the choice of a nonprincipal character (mod 3) for g showsthat the expected estimateXn≤xn≡r (mod D)g(n) =1φ(D)Xn≤x(n,D)=1g(n) + ‘small’is in general false.

In [1, 4, 6] it is shown that the moduli D for which such anestimate fails to be reasonably true are multiples of a single modulus D0. Thepresent situation is arranged so that the complications due to the existence of D0are bound up in the condition (n, P) = 1 and that effectively D0, R have no commondivisors.The moduli dj dividing R, with dj ≤z are dealt with by means of the followingresult.For a multiplicative function g, with values in the complex unit disc, define anexponentially multiplicative function g1 by g1(pk) = g(p)k/k!.

Define the multi-plicative function h by convolution: g = h ∗g1.Thus g(p) = g1(p), h(p) = 0.Moreover, |h(pk)| ≤e. For A ≥0 defineβ1(n) =Xump=nu≤(log x)2Ap≤(log x)6A+15h(u)g1(m)g(p) log plog mp,β2(n) =Xurp=nu≤(log x)2Ar≤(log x)6A+15h(u)g1(r)g(p) log plog rp,and set β(n) = g(n) −β1(n) −β2(n).

Note that βj(n) ≪1 uniformly in n, j.Lemma 1. Let 0 < δ < 1/2.

ThenXD1D2≤xδmax(r,D1D2)=1 maxy≤xXn≤yn≡r (mod D1D2)β(n) −1φ(D)Xn≤y,(n,D2)=1n≡r (mod D1)β(n)≪x(log x)−A(log log x)2 + w−1x(log x)2A+8(log log x)2+ w−1/2x(log x)5/2 log log x,

4P. D. T. A. ELLIOTTwhere D1 is confined to integers whose prime factors do not exceed w and D2 tointegers all of whose prime factors exceed w. The implied constant depends at mostupon δ, A.Lemma 1 represents a generalisation to largely arbitrary multiplicative functionsof the well-known theorem of Bombieri and Vinogradov concerning primes in arith-metic progressions.

The parameter δ may be replaced by 1/2 −ε(x) for a certainpositive function ε(x), which approaches zero as x →∞. Of importance here isthe quality of the error term.

For w ≥(log x)3A+8 it is as good as that of Bombieriand Vinogradov. To this end the functions βj were introduced, manifesting theassertion of [5, p. 408], already in view in [3, p. 178], that for general multiplica-tive functions a change of form would be required.

In particular, β2(n) is largelysupported on the primes and cannot be removed without further information con-cerning g. Most integers n will have few prime divisors, so that effectively the βj(n)are ≪log log x/ log x over the range 2 ≤n ≤x.The functions βj run through the treatment of the integral at (2) along with thecentral function g. A notable feature of the method is the casting of the Selbergsquare functions on the multiplicative integers in a rˆole, which on the additive groupof reals, is traditionally played by a F´ejer kernel. The outcome is the estimate(4)Qh ≪x−1 log wZ 1−1(1 −|t|)e−ithXn≤x(n,P )=1g(n + a) dt + (log x)−1(log log x)2.The complications introduced by the exceptional modulus D0 mentioned earliermust now be dealt with.

To this end [4] or [6] may be applied. For simplicity ofexposition I appeal to Theorem 1 of [6].Lemma 2.

Let 0 < γ < 1, 0 < δ < 1/8, 2 ≤log N ≤Q ≤N.Then anymultiplicative function g with values in the complex unit disc satisfiesXn≤xn≡r (mod D)g(n) =1φ(D)Xn≤x(n,D)=1g(n) + Oxφ(D)log Qlog x1/8−δuniformly for N γ ≤x ≤N, for all (r, D) = 1, for all D ≤Q save possibly for themultiples of a D0 > 1.From Lemma 2 with N = x, Q = exp((log log x)2) I obtain the following esti-mate.Lemma 3. Let w be a power of log x and P the product of the primes in the interval(y, w], where 3|a| ≤y ≤w.

Then eitherXn≤x(n−a,P )=1g(n) =Yyyp −1p −2+ O(x(log x)−1/10),or there is a prime divisor q of P such thatXn≤x(n−a,P )=1g(n) =Yyyp −1p −2+ O(x(log x)−1/10).

ADDITIVE FUNCTIONS ON SHIFTED PRIMES5The implied constants do not depend upon y, g, or q.The prime q may vary with g and x.It follows from (4) and Lemma 3 that(5)Qh ≪x−1Z 1−1(1 −|t|)e−ith Xn≤xg(n)Yp|np>3|a|p −1p −2dt + (log x)−1/12,with possibly a condition (n, q) = 1 required in the sum. Whilst the function g inLemmas 1, 2, and 3 may be arbitrary up to having values in the unit complex disc,in (5) g has the special form exp(itf(n)).

The exceptional prime q may thereforevary with t. It can be arranged that q may only exist on intervals, on each ofwhich it will be constant. The integral at (5) is therefore well defined.

Withoutthe condition (n, q) = 1 we may now follow the original treatment of Ruzsa [9],who considered a similar integral without the weight factor Π(p −1)/(p −2). Theextra condition (n, q) = 1 introduces some further complications, but they can beovercome.SimilarlySh ≪φ(N)−1Z 1−1(1 −|t|)e−ithXn≤N(n,N)=1g(n)Yp|np>3p −1p −2dt + (log N)−1/10.Once again an auxiliary condition (n, q) = 1 may be needed in the sum.

Since Nmay have many prime factors, the condition (n, N) = 1 introduces a new compli-cation, but this, too, can be overcome. It may be remarked here that Theorem 2of [2] shows that in quite general circumstances conditions of the type (n, N) = 1may be factored out of mean values of multiplicative functions.It transpires that the parameter λ appearing in the definitions of W(x) andY (N) may be restricted by |λ| ≤(log x)2, |λ| ≤(log N)2 respectively.References1.

P.D.T.A. Elliott, Multiplicative functions on arithmetic progressions, Mathematika 34(1987), 199–206.2., Extrapolating the mean-values of multiplicative functions, Nederl.

Akad. Weten-sch.

Proc. Ser.

A 92 (1989), 409–420.3., Multiplicative functions on arithmetic progressions III: The large moduli, A trib-ute to Paul Erd¨os (A. Baker, B. Bolob´as, and A. Hajnal, ed. ), Cambridge Univ.

Press,Cambridge, 1990, pp. 177–194.4., Multiplicative functions on arithmetic progressions IV: The middle moduli, J.London Math.

Soc. (2) 41 (1990), 201–216.5., Multiplicative functions on arithmetic progressions V: Composite moduli, J. Lon-don Math.

Soc. (2) 41 (1990), 408–424.6., Multiplicative functions on arithmetic progressions VI: More middle moduli, J.Number Theory (to appear).7.

P. Erd¨os and A. Wintner, Additive arithmetical functions and statistical independence,Amer. J.

Math. 61 (1939), 713–721.8.

G. Hal´asz, On the distribution of additive arithmetical functions, Acta Arith. 27 (1975),143–152.9.

I. Z. Ruzsa, On the concentration of additive functions, Acta Math.

Hungar. 36 (1980),215–232.

6P. D. T. A. ELLIOTT10.

N. M. Timofeev, The Erd¨os-Kubilius conjecture concerning the value distribution of ad-ditive functions on the seqeunce of shifted primes, Acta Arith. LVIII (1991), 113–131.

(Russian)Department of Mathematics, University of Colorado, Boulder, Colorado 80309-0001

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