Criteria for irrationality of Euler’s constant

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 $n>0$, we define a double integral $I_n$ and a positive integer $S_n$, 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 $n$. 2. The formula holds for all sufficiently large $n$. 3. Euler’s constant is a rational number. A corollary is that if $\{\log S_n\}\ge 2^{-n}$ infinitely often, then $\gamma$ is irrational. Indeed, if the inequality holds for a given $n$ (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.