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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Criteria for irrationality of Euler’s constant
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by Jonathan Sondow PDF
Proc. Amer. Math. Soc. 131 (2003), 3335-3344 Request permission

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.
References
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Additional Information
  • Jonathan Sondow
  • Affiliation: 209 West 97th Street, New York, New York 10025
  • Email: jsondow@alumni.princeton.edu
  • Received by editor(s): June 4, 2002
  • Published electronically: March 11, 2003
  • Communicated by: David E. Rohrlich
  • © Copyright 2003 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 131 (2003), 3335-3344
  • MSC (2000): Primary 11J72; Secondary 05A19
  • DOI: https://doi.org/10.1090/S0002-9939-03-07081-3
  • MathSciNet review: 1990621