Criteria for Irrationality of Euler's Constant
| dc.creator | Sondow, Jonathan | |
| dc.date | 2002-09-06 | |
| dc.date | 2002-10-04 | |
| dc.date.accessioned | 2026-07-07T04:50:40Z | |
| dc.date.available | 2026-07-07T04:50:40Z | |
| dc.description | 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)} > 1/2^n 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. | |
| dc.description | 12 pages, 1 figure, 2 tables, proofs shortened & typos fixed, revised version accepted by Proc. Amer. Math. Soc | |
| dc.identifier | https://arxiv.org/abs/math/0209070 | |
| dc.identifier | http://arxiv.org/abs/math/0209070 | |
| dc.identifier | Proc. Amer. Math. Soc. 131 (2003) 3335-3344 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/64873 | |
| dc.subject | Number Theory | |
| dc.subject | Combinatorics | |
| dc.subject | Primary 11J72, Secondary 05A19 | |
| dc.title | Criteria for Irrationality of Euler's Constant | |
| dc.type | text |