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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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A fast algorithm to compute the Ramanujan-Deninger gamma function and some number-theoretic applications
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by Alessandro Languasco and Luca Righi HTML | PDF
Math. Comp. 90 (2021), 2899-2921 Request permission

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.

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Additional Information
  • Alessandro Languasco
  • Affiliation: Università di Padova, Dipartimento di Matematica “Tullio Levi-Civita”, Via Trieste 63, 35121 Padova, Italy
  • MR Author ID: 354780
  • ORCID: 0000-0003-2723-554X
  • Email: alessandro.languasco@unipd.it
  • Luca Righi
  • Affiliation: Università di Padova, Dipartimento di Matematica “Tullio Levi-Civita”, Via Trieste 63, 35121 Padova, Italy
  • ORCID: 0000-0002-4522-7511
  • Email: righi@math.unipd.it
  • Received by editor(s): May 24, 2020
  • Received by editor(s) in revised form: February 20, 2021
  • Published electronically: August 12, 2021
  • Additional Notes: The calculations here described in Section \ref{sect-implementation} were performed using the University of Padova Strategic Research Infrastructure Grant 2017: “CAPRI: Calcolo ad Alte Prestazioni per la Ricerca e l’Innovazione”, http://capri.dei.unipd.it.
    The first author is the corresponding author
  • © Copyright 2021 American Mathematical Society
  • Journal: Math. Comp. 90 (2021), 2899-2921
  • MSC (2020): Primary 33-04, 11-04; Secondary 33E20, 11Y16, 11Y60
  • DOI: https://doi.org/10.1090/mcom/3668
  • MathSciNet review: 4305373