# On the possible exceptions for the transcendence of the log-gamma function at rational entries - Mathematics > Number Theory

On the possible exceptions for the transcendence of the log-gamma function at rational entries - Mathematics > Number Theory - Descarga este documento en PDF. Documentación en PDF para descargar gratis. Disponible también para leer online.

Abstract: In a recent work JNT \textbf{129}, 2154 2009, Gun and co-workers haveclaimed that the number $\,\log{\Gammax} + \log{\Gamma1-x}\,$, $x$ being arational number between $0$ and $1$, is transcendental with at most \emph{one}possible exception, but the proof presented there in that work is\emph{incorrect}. Here in this paper, I point out the mistake they committedand I present a theorem that establishes the transcendence of those numberswith at most \emph{two} possible exceptions. As a consequence, I make use ofthe reflection property of this function to establish a criteria for thetranscendence of $\,\log{\pi}$, a number whose irrationality is not proved yet.This has an interesting consequence for the transcendence of the product $\,\pi\cdot e$, another number whose irrationality remains unproven.

Autor: F. M. S. Lima

Fuente: https://arxiv.org/