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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.

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