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Criteria for irrationality of Euler's constant

Authors:	Jonathan Sondow

Institution:	209 West 97th Street, New York, New York 10025

Abstract:	By modifying Beukers' proof of Apéry's theorem that $\zeta(3)$ is irrational, we derive criteria for irrationality of Euler's constant, $\gamma$ . For , we define a double integral and a positive integer , and prove that with $d_n=\operatorname{LCM}(1,\dotsc,n)$ the following are equivalent: 1. The fractional part of $\log S_n$ is given by $\{\log S_n\}=d_{2n}I_n$ for some . 2. The formula holds for all sufficiently large . 3. Euler's constant is a rational number. A corollary is that if $\{\log S_n\}\ge 2^{-n}$ infinitely often, then $\gamma$ is irrational. Indeed, if the inequality holds for a given (we present numerical evidence for $1\le n\le 2500)$ and $\gamma$ is rational, then its denominator does not divide $d_{2n}\binom{2n}{n}$ . We prove 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.

Keywords:	Irrationality Euler's constant Ap\'ery's theorem Beukers' integrals linear form in logarithms fractional part harmonic number Prime Number Theorem Laplace's method asymptotic formula combinatorial identity

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