2002/09/30 by Jonathan Sondow
Mathematics · #math.NT #math.CO #msc:11J72 #msc:05A19
published as Proc. Amer. Math. Soc. 131 (2003) 3335-3344 · 12 pages, 1 figure, 2 tables, proofs shortened & typos fixed, revised version accepted by Proc. Amer. Math. Soc
arxiv created 2002/10/04 · arxiv updated 2009/11/30
By modifying Beukers' proof of Apery'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 if d(n) = LCM(1,...,n), then the fractional part of logS(n) is given by logS(n) = d(2n)I(n), for all n sufficiently large, if and only if gamma is a rational number. A corollary is that if logS(n) > 1/2n infinitely often, then gamma is irrational. Indeed, if the inequality holds for a given n (we present numerical evidence for 0 < n < 2500 and n = 10000) and gamma is rational, then its denominator does not divide the product d(2n)Binomial(2n,n). We prove a new combinatorial identity in order to show that a certain linear form in logarithms is in fact logS(n). A by-product is a rapidly converging asymptotic formula for gamma, used by P. Sebah to compute it correct to 18063 decimals.