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


References [Enhancements On Off] (What's this?)

  • 1. R. Apéry, Irrationalité de $\zeta(2)$ et $\zeta(3)$, Astérisque 61 (1979), 12-14.
  • 2. F. Beukers, A note on the irrationality of $\zeta(2)$ and $\zeta(3)$, Bull. London Math. Soc. 12 (1979), 268-272. MR 81j:10045
  • 3. N. Bleistein and R. Handelsman, Asymptotic expansion of integrals, Holt, Rinehart and Winston, 1975. MR 89d:41049 (review of 2nd edition)
  • 4. K. Ball and T. Rivoal, Irrationalité d'une infinité de valeurs de la fonction zêta aux entiers impairs, Invent. Math. 156 (2001), 193-207. MR 2003a:11086
  • 5. D. Huylebrouck, Similarities in irrationality proofs for $\pi,\ln 2, \zeta(2)$, and $\zeta(3)$, Amer. Math. Monthly 118 (2001), 222-231. MR 2002b:11095
  • 6. Y. Nesterenko, A few remarks on $\zeta(3)$, Math. Notes 59 (1996), 625-636. MR 98b:11088
  • 7. A. van der Poorten, A proof that Euler missed$\dotsc$Apéry's proof of the irrationality of $\zeta(3)$, Math. Intelligencer 1 (1979), 195-203. MR 80i:10054
  • 8. J. B. Rosser and L. Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. of Math. 6 (1962), 64-94. MR 25:1139
  • 9. P. Sebah, personal communication, 30 July 2002.
  • 10. J. Sondow, Hypergeometric and double integrals for Euler's constant, Amer. Math. Monthly, submitted.
  • 11. -, A hypergeometric approach, via linear forms involving logarithms, to irrationality criteria for Euler's constant, CRM Conference Proceedings of CNTA 7 (May, 2002), to appear.
  • 12. -, An irrationality measure for Liouville numbers and conditional measures for Euler's constant, in preparation.
  • 13. W. Zudilin, One of the numbers $\zeta(5), \zeta(7), \zeta(9), \zeta(11)$ is irrational, Russian Math. Surveys 56:4 (2001), 774-776. MR 2002g:11090

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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

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