Dedekind Sums, Mean Square Value of $L$-Functions at $s=1$ and Upper Bounds on Relative Class Numbers

Volume 64 / 2016

Stéphane R. Louboutin Bulletin Polish Acad. Sci. Math. 64 (2016), 165-174 MSC: Primary 11R42; Secondary 11M20, 11R20, 11R29. DOI: 10.4064/ba8092-12-2016 Published online: 14 December 2016

Abstract

Explicit formulas for the quadratic mean value at $s=1$ of the Dirichlet $L$-functions associated with the set $X_f^-$ of the $\phi (f)/2$ odd Dirichlet characters mod $f$ are known. They have been used to obtain explicit upper bounds for relative class numbers of cyclotomic number fields. Here we present a generalization of these results: we show that explicit formulas for quadratic mean values at $s=1$ of Dirichlet $L$-functions associated with subsets of $X_f^-$ can be obtained. As an application we use them to obtain explicit upper bounds for relative class numbers of imaginary subfields of cyclotomic number fields.

Authors

  • Stéphane R. LouboutinAix Marseille Université CNRS, Centrale Marseille, I2M
    Marseille, France
    e-mail

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