Points rationnels de la fonction Gamma d'Euler

TL;DR

This paper establishes a new zero estimate for polynomials involving the Gamma function, leading to bounds on the number of rational points with rational Gamma values within specific intervals.

Contribution

It introduces a novel method adapted from the Riemann zeta-function to analyze rational values of the Gamma function, providing effective bounds on such points.

Findings

Bound on the number of rational points with rational Gamma values

Effective constants for intervals [n-1,n]

Logarithmic bounds on denominators

Abstract

We use a method, first developed for the Riemann zeta-function by Masser in ["Rational values of the Riemann zeta function", Journ. Num. Th. 131 (2011), 2037-2046], to prove a new zero estimate for polynomials in z and 1/Gamma(z). This allows us to prove that, for all n>=2, there exists an absolute effective positive constant C(n) such that, for all D>=3, there are at most C(n)log^2(D)/loglog(D) rational numbers z in [n-1,n] with denominator at most D and such that Gamma(z) is also rational with denominator at most D.