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 $M_q={\max }_{\chi \ne {\chi }_0}|L^{\prime }/L(1,\chi )|$, where $q$ is an odd prime, $\chi$ runs over the primitive Dirichlet characters $\mathrm{mod}q$, ${\chi }_0$ is the trivial Dirichlet character $\mathrm{mod}q$ and $L(s,\chi )$ is the Dirichlet $L$-function associated to $\chi$. Using such algorithms we obtained that ${\mathfrak{G}}_{50040955631}=-0.16595399\dots$ and ${\mathfrak{G}}_{50040955631}^{+}=13.89764738\dots$ 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 $q$, $10^6<q\le 10^7$, thus extending previous results. 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 odd 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 \url{http://www.math.unipd.it/~languasc/Scomp-appl.html}.

to appear in Math. Comp. Inserted section 5.7 and two histograms. Several typos corrected

References in corpus (1)

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

Cited by in corpus (8)

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• Numerical estimates on the Landau-Siegel zero and other related quantities
• The Kummer ratio of the relative class number for prime cyclotomic fields
• Euler constants from primes in arithmetic progression
• Relative class numbers and Euler-Kronecker constants of maximal real cyclotomic subfields