Transcendence of the log gamma function and some discrete periods
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].
Baker's Theorem on linear independence of logarithms of algebraic numbers
scientific article; zbMATH DE number 5575282