Criteria for irrationality of Euler's constant
Author:
Jonathan Sondow
Journal:
Proc. Amer. Math. Soc. 131 (2003), 33353344
MSC (2000):
Primary 11J72; Secondary 05A19
Published electronically:
March 11, 2003
MathSciNet review:
1990621
Fulltext 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 byproduct is a rapidly converging asymptotic formula for , used by P. Sebah to compute correct to 18063 decimals.
 1.
R. Apéry, Irrationalité de et , Astérisque 61 (1979), 1214.
 2.
F.
Beukers, A note on the irrationality of 𝜁(2) and
𝜁(3), Bull. London Math. Soc. 11 (1979),
no. 3, 268–272. MR 554391
(81j:10045), http://dx.doi.org/10.1112/blms/11.3.268
 3.
Norman
Bleistein and Richard
A. Handelsman, Asymptotic expansions of integrals, 2nd ed.,
Dover Publications, Inc., New York, 1986. MR 863284
(89d:41049)
 4.
Keith
Ball and Tanguy
Rivoal, Irrationalité d’une infinité de valeurs
de la fonction zêta aux entiers impairs, Invent. Math.
146 (2001), no. 1, 193–207 (French). MR 1859021
(2003a:11086), http://dx.doi.org/10.1007/s002220100168
 5.
Dirk
Huylebrouck, Similarities in irrationality proofs for 𝜋,
ln2, 𝜁(2), and 𝜁(3), Amer. Math. Monthly
108 (2001), no. 3, 222–231. MR 1834702
(2002b:11095), http://dx.doi.org/10.2307/2695383
 6.
Yu.
V. Nesterenko, Some remarks on 𝜁(3), Mat. Zametki
59 (1996), no. 6, 865–880, 960 (Russian, with
Russian summary); English transl., Math. Notes 59 (1996),
no. 56, 625–636. MR 1445472
(98b:11088), http://dx.doi.org/10.1007/BF02307212
 7.
Alfred
van der Poorten, A proof that Euler
missed…Apéry’s proof of the irrationality of
𝜁(3), Math. Intelligencer 1 (1978/79),
no. 4, 195–203. An informal report. MR 547748
(80i:10054), http://dx.doi.org/10.1007/BF03028234
 8.
J.
Barkley Rosser and Lowell
Schoenfeld, Approximate formulas for some functions of prime
numbers, Illinois J. Math. 6 (1962), 64–94. MR 0137689
(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.
Noam
D. Elkies and Benedict
H. Gross, Cubic rings and the exceptional Jordan algebra, Duke
Math. J. 109 (2001), no. 2, 383–409. MR 1845183
(2002g:11090), http://dx.doi.org/10.1215/S0012709401109241
 1.
 R. Apéry, Irrationalité de et , Astérisque 61 (1979), 1214.
 2.
 F. Beukers, A note on the irrationality of and , Bull. London Math. Soc. 12 (1979), 268272. 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), 193207. MR 2003a:11086
 5.
 D. Huylebrouck, Similarities in irrationality proofs for , and , Amer. Math. Monthly 118 (2001), 222231. MR 2002b:11095
 6.
 Y. Nesterenko, A few remarks on , Math. Notes 59 (1996), 625636. MR 98b:11088
 7.
 A. van der Poorten, A proof that Euler missedApéry's proof of the irrationality of , Math. Intelligencer 1 (1979), 195203. MR 80i:10054
 8.
 J. B. Rosser and L. Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. of Math. 6 (1962), 6494. 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 is irrational, Russian Math. Surveys 56:4 (2001), 774776. MR 2002g:11090
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2000):
11J72,
05A19
Retrieve articles in all journals
with MSC (2000):
11J72,
05A19
Additional Information
Jonathan Sondow
Affiliation:
209 West 97th Street, New York, New York 10025
Email:
jsondow@alumni.princeton.edu
DOI:
http://dx.doi.org/10.1090/S0002993903070813
PII:
S 00029939(03)070813
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
