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 principal 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 prime $q$, $10^6< q\le {10^{7}}$, thus extending the results by Languasco [Res. Number Theory 7 (2021), no. 1, Paper No. 2]. As a consequence we obtain that both $\mathfrak {G}_q$ and $\mathfrak {G}_q^+$ are positive for every odd prime $q$ up to ${10^{7}}$ and that $\frac {17}{20} \log \log q< M_q < \frac {5}{4} \log \log q$ for every prime $1531 < q\le {10^{7}}$. In fact the lower bound holds true for $q>13$. The programs used and the results here described are collected at the following address http://www.math.unipd.it/ languasc/Scomp-appl.html.