Abstract
AbstractWe give a new hypergeometric construction of rational approximations to ζ(4), which absorbs the earlier one from 2003 based on Bailey's 9F8 hypergeometric integrals. With the novel ingredients we are able to gain better control of the arithmetic and produce a record irrationality measure for ζ(4).
Highlights
Apery’s proof [1, 6, 18] of the irrationality of ζ(3) in the 1970s sparked research in arithmetic on the values of Riemann’s zeta function ζ(s) at integers s ≥ 2
Some particular representatives of this development include [4, 8, 9, 13], and the story culminated in a remarkable arithmetic method [14, 15] of Rhin and Viola to produce sharp irrationality measures for ζ(2) and ζ(3) using groups of transformations of rational approximations to the quantities
The hypergeometric machinery has proven to be useful in further arithmetic applications; see, for example, [7, 11, 12, 22] for more recent achievements
Summary
Apery’s proof [1, 6, 18] of the irrationality of ζ(3) in the 1970s sparked research in arithmetic on the values of Riemann’s zeta function ζ(s) at integers s ≥ 2. The principal goal of this work is to recast the rational approximations to ζ(4) from [19] in a different (but still hypergeometric) form and obtain, by these means, better control of the arithmetic of their coefficients. In this way, we are able to produce the estimate μ(ζ(4)) 12.51085940. For the irrationality exponent of the zeta value, which is better than the conjectural one given in [19] This is not surprising, as we do not attempt to prove the denominator conjecture from [19] but instead investigate the arithmetic of approximations from a different hypergeometric family. × Γ(1 + h0 − h−1 + s + t) sin π(s + t) ds dt
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