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Criteria for irrationality of Euler's constant

Author: Jonathan Sondow
Journal: Proc. Amer. Math. Soc. 131 (2003), 3335-3344
MSC (2000): Primary 11J72; Secondary 05A19
Published electronically: March 11, 2003
MathSciNet review: 1990621
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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.

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Additional Information

Jonathan Sondow
Affiliation: 209 West 97th Street, New York, New York 10025

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
Received by editor(s): June 4, 2002
Published electronically: March 11, 2003
Communicated by: David E. Rohrlich
Article copyright: © Copyright 2003 American Mathematical Society