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

Abstract

Let $L(s,\chi )$ be the Dirichlet $L$-function associated to a non trivial primitive Dirichlet character $\chi$ defined $\mathrm{mod}q$, where $q$ is an odd prime. In this paper we introduce a fast method to compute $|L(1,\chi )|$ 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 $|L(1,\chi )|$, where $\chi$ runs over the non trivial primitive Dirichlet characters $\mathrm{mod}q$, 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}.

21 pages, 2 tables, 12 figures. One reference updated