Transcendence of the log gamma function and some discrete periods
type: Item
instance of: scholarly article
title: Transcendence of the log gamma function and some discrete periods
zbMATH Open document ID: 1175.11039
author: Sanoli Gun
author: M. Ram Murty
author: Purusottam Rath
published in: Journal of Number Theory
publication date: 2009-07-06T00:00:00Z
review text: Using the reflection formula \(\Gamma(z)\Gamma(1-z)=\pi/\sin(\pi z)\) together with Baker's Theorem on linear independence of logarithms of algebraic numbers, the authors deduce that there is at most one rational number \(x\) in the interval \(0<x\leq 1/2\) for which \(\log\Gamma(x)+\log\Gamma(1-x)\) is algebraic. Equivalently, in the set of numbers \(\log\Gamma(x)+\log\Gamma(1-x)\) where \(x\) runs over the rational numbers in the interval \(0<x<1\), there is at most one algebraic number. The authors deduce further similar statements from Schanuel's Conjecture. They also investigate the Diophantine nature of series like \(\sum_n P(n)/Q(n)\) when \(P\) and \(Q\) are polynomials with rational coefficients, and \(n \) runs either over the positive integers or else over the rational integers, avoiding the poles of \(P/Q\). They deduce their results from the theorem of Lindemann--Weierstrass, from the theorem of Nesterenko, or from Schanuel's Conjecture. A corrigendum concerning a corrected proof is in print [see doi:10.1016/j.jnt.2009.11.002].
reviewed by: Michel Waldschmidt
Mathematics Subject Classification ID: 11J81
Mathematics Subject Classification ID: 11J86
Mathematics Subject Classification ID: 11J91
zbMATH DE Number: 5575282
zbMATH Keywords: Baker's Theorem on linear independence of logarithms of algebraic numbers
zbMATH Keywords: Schanuel's conjecture
zbMATH Keywords: Nesterenko's Theorem
zbMATH Keywords: transcendence
zbMATH Keywords: Gamma function
MaRDI profile type: Publication
full work available at URL: j.jnt.2009.01.008
OpenAlex ID: W2041101352
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DOI: 10.1016/J.JNT.2009.01.008
description: scientific article; zbMATH DE number 5575282