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

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 $G_q$, $G_q^+$ and $M_q=max_{chi e chi_0} ert L^prime/L(1,chi) ert$, where $q$ is an odd prime, $chi$ runs over the primitive Dirichlet characters $mod q$, $chi_0$ is the principal Dirichlet character $mod q$ and $L(s,chi)$ is the Dirichlet $L$-function associated to $chi$. Using such algorithms we obtained that $G_{50 040 955 631} =-0.16595399dotsc$ and $G_{50 040 955 631}^+ =13.89764738dotsc$ thus getting a new negative value for $G_q$. Moreover we also computed $G_q$, $G_q^+$ and $M_q$ for every odd prime $q$, $10^6< q le 10^7$, thus extending the results in Languasco (2019). As a consequence we obtain that both $G_q$ and $G_q^+$ are positive for every odd prime $q$ up to $10^7$ and that $rac{17}{20} log log q< M_q < rac{5}{4} log log q $ for every odd prime $1531 < qle 10^7$. In fact the lower bound holds true for $q>13$. The programs used and the results here described are collected at the following address \url{http://www.math.unipd.it/~languasc/Scomp-appl.html}.