Arithmetic of Catalan’s constant and its relatives

We prove that at least one of the six numbers β(2i)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta (2i)$$\end{document} for i=1,…,6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i=1,\ldots ,6$$\end{document} is irrational. Here β(s)=∑k=0∞(-1)k(2k+1)-s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta (s)=\sum _{k=0}^{\infty }(-1)^k(2k+1)^{-s}$$\end{document} denotes Dirichlet’s beta function, so that β(2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta (2)$$\end{document} is Catalan’s constant.


Introduction
In this note we discuss arithmetic properties of the values of Dirichlet's beta function β(s) = ∞ n=1 −4 n n s = ∞ k=0 (−1) k (2k + 1) s at positive even integers s. The very first such beta value β (2) is famously known as Catalan's constant; its irrationality remains an open problem, though we expect the number to be irrational and transcendental. The best known results in this direction were given by T. Rivoal and this author in [4]. Namely, we showed that at least one of the seven numbers β (2), β (4), . . . , β(14) is irrational, and that there are infinitely many irrational numbers among the even beta values β (2), β (4), β (6), . . . . Here we use a variant of the method from [3,8] to improve slightly on the former achievement; a significant strengthening towards the infinitude result, based on a further development of the ideas in [2,6], is a subject of the recent preprint [1] of S. Fischler.
In Sect. 2 we illustrate principal ingredients of the method in a particularly simple situation; this leads to a weaker version of Theorem 1, namely, to the irrationality of at least one number β(2i) for i = 1, . . . , 8. The details about the general construction of approximating forms to even beta values and our proof of Theorem 1 are given in Sect. 3.

Outline of the construction
For an odd integer s ≥ 3 (which we eventually set to be 17) and even n > 0, define the rational function (1) and assign to it the related sequence of quantities The sums r n are instances of generalized hypergeometric functions, for which we can use some standard integral representations to write (details are given in Lemma 2 below). This form clearly implies that r n > 0 and also gives access to the asymptotics lim n→∞ r 1/n n = 2 6 3 3 max An important ingredient of the construction is the following decomposition of the quantities r n .

Lemma 1
For odd s and even n as above, where a i = a i,n satisfy the inclusions −1 n d s−i n a i ∈ Z for i = 0, 1, . . . , s even. Here d n denotes the least common multiple of the numbers 1, 2, . . . , n, and n = 2 √ n< p≤n the function ψ(x) denotes the logarithmic derivative of the gamma function.

Remark 1
The analogous construction in [4] makes use of a slightly different rational function than (1), namely, of The analogous decomposition of a related quantity r n assumes the form in which the rational coefficients a i = a i,n satisfy −1 n d s−i n a i ∈ Z for i = 1, . . . , s even, but −1 n d s 2n a 0 ∈ Z. The appearance of d s 2n as the common denominator in place of d s n changes the scene drastically and leads to weaker arithmetic applications.

Proof of Lemma 1
Following the strategy in [5,8] we can write the function (1) as sum of partial fractions, 2n k t + k and 2t + n; the inclusions d s−i n a i,k ∈ Z then follow from [8, Lemma 1]. The cancellation by the factor n originates from the p-adic analysis of the binomial factors entering is a periodic function of period 1 in both x and y, and from the inequality the details can be borrowed from [4,Sect. 7]. Furthermore, the property R n (−t − n) = R n (t) derived from (1) implies a i,k = (−1) i a i,n−k for i = 1, . . . , s and k = 0, 1, . . . , n.
Recall that n is even, so that n/2 = m is a positive integer. The summation over ν in (2) can also start from −m − 1 (rather than 1 or n + 1), because the function R n (t) vanishes at all half-integers between −2n and n. Therefore, Finally,
Define the rational function where , and the (very-well-poised) hypergeometric sum Then [4, Lemma 1] implies the following Euler-type integral representation of r n (see also [4,Lemma 3]).

Lemma 2 The formula
is valid. In particular, r n > 0 and Computation of the latter maximum is performed in [4,Sect. 4,Remark], and the result is as follows.

Lemma 3
Assume that x 0 is a unique zero of the polynomial in the interval 0 < x < 1, and set Arithmetic ingredients of the construction are in line with the strategy used in the proof of Lemma 1. For simplicity we split the corresponding statement into two parts. Define and notice that the poles of the rational function (4) are located at the points t = −k − 1 2 for integers k in the range N ≤ k ≤ h 0 − N − 1.

Lemma 4 The coefficients in the partial-fraction decomposition
and Proof For this, we write the function R n (t − 1 2 ) as the product of 2t + h 0 − 1, the three integer-valued polynomials This means that the positive linear forms −1 n d 13 M r n ∈ Zβ(2) + Zβ(4) + · · · + Zβ(12) + Z tend to 0 as n → ∞. Thus, at least one of the even beta values in consideration must be irrational.
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