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Criteria for irrationality of Euler’s constant. (English) Zbl 1113.11040
By modifying Beukers’ proof of Apéry’s theorem that \(\zeta(3)\) is irrational, the author derives criteria for irrationality of Euler’s constant, \(\gamma\). For \(n>0\), define a double integral \(I_n\) and a positive integer \(S_n\), then it is proved that with \(d_n=\text{lcm}(1,\dots,n)\) the following are equivalent:
1) The fractional part of \(\log S_n\) is given by \(\{\log S_n\}=d_{2n}I_n\) for some \(n\).
2) The formula holds for all sufficiently large \(n\).
3) Euler’s constant is a rational number.
A corollary is that if \(\{\log S_n\}\geq 2^{-n}\) infinitely often, then \(\gamma\) is irrational. Indeed, if the inequality holds for a given \(n\) (numerical evidence for \(1\leq n\leq 2500\) is given by the author) and \(\gamma\) is rational, then its denominator does not divide \(d_{2n}\binom{2n}{n}\). The author also proves a new combinatorial identity in order to show that a certain linear form in logarithms is in fact \(\log S_n\). A by-product is a rapidly converging asymptotic formula for \(\gamma\), used by P. Sebah to compute \(\gamma\) correct to 18063 decimals.

11J72 Irrationality; linear independence over a field
05A19 Combinatorial identities, bijective combinatorics
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