Numerical verification of Littlewood's bounds for $\vert L(1,\chi)\vert$

Abstract

Let $L(s,\chi)$ be the Dirichlet $L$-function associated to a non trivial primitive Dirichlet character $\chi$ defined $\bmod\ q$, where $q$ is an odd prime. In this paper we introduce a fast method to compute $\vert L(1,\chi) \vert$ using the values of Euler's $\Gamma$ function. We also introduce an alternative way of computing $\log \Gamma(x)$ and $\psi(x)= \Gamma^\prime/\Gamma(x)$,$x\in(0,1)$. Using such algorithms we numerically verify the classical Littlewood bounds and the recent Lamzouri-Li-Soundararajan estimates on $\vert L(1,\chi) \vert$, where $\chi$ runs over the non trivial primitive Dirichlet characters $\bmod\ q$, for every odd prime $q$ up to $10^7$. The programs used and the results here described are collected at the following address \url{http://www.math.unipd.it/~languasc/Littlewood_ineq.html}.