Approximate expressions for mathematical constants from PSLQ algorithm: a simple approach and a case study

For positive integer values of s, s > 1, Euler was the first to derive (in 1734) an exact closed-form expression for ζ(s), namely ζ(2) = π2 /6, the solution of the Basel problem (see Ref. [2] and references therein). Some years later (the result was found in 1739, but published only in 1750), he succeeded in extending his result to all even values of s [1]:

where n is a positive integer and B 2n ∈ Q are Bernoulli numbers. For odd values of s, however, no exact closed-form expression is currently known. 2 The increase of interest in ζ(3), which comes from both pure and applied mathematics,3 has stimulated its high-precision numerical computation [18], as well as the search for simple approximate expressions [24]. Let us adopt a reasonable criterium for the adjective “simple,” in the context of finite approximate expressions. Here, it will designate closed-form expressions containing a few terms/factors composed by other known mathematical constants and a few integer numbers with small absolute value. This criterium is, of course, vague due to the forms “a few” and “small”. To make it not-so-imprecise, “a few” will mean less than, say 10, and “small” will mean less than, say 500. A such approximation was presented by Galliani (2002), namely [23]

where γ is the Euler-Mascheroni constant, which is accurate to 4 decimal places.

Another nice simple approximation is

due to Hudson (2004), which is accurate to 7 places [23]. Among the many approximations for ζ(3) presented by Hudson, the most accurate is [23]

1 The partial sum N n=1 1/n 3 , N being any positive integer, of course yields a rational approximation to ζ(3). However, even for N as large as 100 yields only four correct decimal places.

which yields 12 correct decimal places and, clearly, is not a simple approximate expression (in our terminology). The same for ζ(3) ≈ 97525 2515594 π 3 , which I have found on searching for a direct integer relation between ζ(3) and π 3 . On trying to reach greater accuracy, however, one soon observes that it is very difficult to avoid the appearance of large integers.

In this note, I show how to use PSLQ algorithm for finding the best rational coefficients, within a desired precision, of a null linear combination of a finite number of real constants. As an example, I explore approximate expressions for the Apéry’s constant ζ(3), a number to which no closed-form expression in terms of a finite combination of elementary functions of known constants is known. For this, I choose a suitable search basis composed by numbers which seem to be closely related to ζ(3), namely π, ln 2 , ln (1 + √ 2 ), and G (the Catalan’s constant). This yields a simple expression for ζ(3) accurate to 21 decimal places. The short Maple TM code used in this computation is included to stimulate the readers to conduct their own experiments.

An important task in experimental mathematics is to search for integer relations involving a finite set of known numbers. An integer relation algorithm is a computational scheme that, for a given real vector x = (x 1 , x 2 , . . . , x n ), n > 1, it either finds a nonnull vector of integers a = (a 1 , a 2 , . . . , a n ) such that a 1 x 1 + a 2 x 2 + . . . + a n x n = 0 or else establishes that there is no such integer vector within a ball of some radius about the origin. 4Presently, the best algorithm for detecting integer relations is the PSLQ algorithm (acronym for Partial Sum of Least sQuares) introduced by Ferguson and Bailey (1992) [13]. A simplified formulation for this algorithm, equivalent to the original one, was subsequently developed by Ferguson and coworkers (1999) [14]. This more efficient version is currently implemented in both Maple TM and Mathematica TM , two of the most popular mathematical softwares. This version of PSLQ, optimized with certain reduction schemes, was named one of the ’ten algorithms of the century’ in Ref. [3].

In short, PSLQ operates as follows. Given a vector x of n given real numbers, input as a list of floating-point (FP) numbers, the algorithm uses QR decomposition in order to construct a series of matrices A m such that the absolute values of the entries of the vector y m = A -1 m • x decrease monotonically. At any given iteration, the largest and smallest entries of y m usually differ by no more than a few orders of magnitude. When the desired integer relation is detected, the smalle

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