Criteria for irrationality of Euler’s constant

Abstract

By modifying Beukers’ proof of Apéry’s theorem that ζ ( 3 ) \zeta (3) is irrational, we derive criteria for irrationality of Euler’s constant, γ \gamma . For n > 0 n>0 , we define a double integral I n I_n and a positive integer S n S_n , and prove that with d n = LCM ⁡ ( 1 , … , n ) d_n=\operatorname {LCM}(1,\dotsc ,n) the following are equivalent: 1. The fractional part of log ⁡ S n \log S_n is given by { log ⁡ S n } = d 2 n I n \{\log S_n\}=d_{2n}I_n for some n n . 2. The formula holds for all sufficiently large n n . 3. Euler’s constant is a rational number. A corollary is that if { log ⁡ S n } ≥ 2 − n \{\log S_n\}\ge 2^{-n} infinitely often, then γ \gamma is irrational. Indeed, if the inequality holds for a given n n (we present numerical evidence for 1 ≤ n ≤ 2500 ) 1\le n\le 2500) and γ \gamma is rational, then its denominator does not divide d 2 n ( 2 n n ) 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 \log S_n . A by-product is a rapidly converging asymptotic formula for γ \gamma , used by P. Sebah to compute γ \gamma correct to 18063 decimals.