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 allows us greatly extend numerical investigations about Euler-Kronecker constants $\mathfrak{G}_q$, $\mathfrak{G}_q^+$ $M_q=\max_{\chi\ne \chi_0} \vert L^\prime/L(1,\chi)\vert$, where $q$ is odd prime, $\chi$ runs over primitive Dirichlet characters $\bmod\ q$, $\chi_0$ trivial character q$ $L(s,\chi)$ $L$-function associated $\chi$. Using such algorithms we obtained that $\mathfrak{G}_{50 040 955 631} =-0.16595399\dotsc$ 631}^+ =13.89764738\dotsc$ thus getting new negative value for $\mathfrak{G}_q$. Moreover also computed $M_q$ every prime $q$, $10^6< q\le 10^7$, extending previous results. As consequence obtain both $\mathfrak{G}_q$ are up $10^7$ $\frac{17}{20} \log q< M_q < \frac{5}{4} q $ $1531 10^7$. In fact lower bound holds true $q>13$. The programs used results here described collected following address \url{http://www.math.unipd.it/~languasc/Scomp-appl.html}.