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

DOI:
https://doi.org/10.1090/S0002-9939-03-07081-3

Published electronically:
March 11, 2003

MathSciNet review:
1990621

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: By modifying Beukers' proof of Apéry's theorem that is irrational, we derive criteria for irrationality of Euler's constant, . For , we define a double integral and a positive integer , and prove that with the following are equivalent:

1. The fractional part of is given by for some .

2. The formula holds for all sufficiently large .

3. Euler's constant is a rational number.

A corollary is that if infinitely often, then is irrational. Indeed, if the inequality holds for a given (we present numerical evidence for and is rational, then its denominator does not divide . We prove a new combinatorial identity in order to show that a certain linear form in logarithms is in fact . A by-product is a rapidly converging asymptotic formula for , used by P. Sebah to compute correct to 18063 decimals.

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Additional Information

**Jonathan Sondow**

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

Email:
jsondow@alumni.princeton.edu

DOI:
https://doi.org/10.1090/S0002-9939-03-07081-3

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