A fast algorithm to compute the Ramanujan-Deninger gamma function and some number-theoretic applications

Abstract

We introduce a fast algorithm to compute the Ramanujan-Deninger gamma function and its logarithmic derivative at positive values. Such an algorithm allows us to greatly extend the numerical investigations about the Euler-Kronecker constants $\\mathfrak{G}_q$, $\\mathfrak{G}_q^+$ and $M_q=\\max_{\\chi\\ne \\chi_0} \\vert L^\\prime/L(1,\\chi)\\vert$, where $q$ is an odd prime, $\\chi$ runs over the primitive Dirichlet characters $\\bmod\\ q$, $\\chi_0$ is the trivial Dirichlet character $\\bmod\\ q$ and $L(s,\\chi)$ is the Dirichlet $L$-function associated to $\\chi$. Using such algorithms we obtained that $\\mathfrak{G}_{50 040 955 631} =-0.16595399\\dotsc$ and $\\mathfrak{G}_{50 040 955 631}^+ =13.89764738\\dotsc$ thus getting a new negative value for $\\mathfrak{G}_q$. Moreover we also computed $\\mathfrak{G}_q$, $\\mathfrak{G}_q^+$ and $M_q$ for every odd prime $q$, $10^6 13$. The programs used and the results here described are collected at the following address \\url{this http URL}.