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

Stéphane R. Louboutin¹

¹ Aix Marseille Université CNRS, Centrale Marseille, I2M Marseille, France

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 64 (2016) no. 2-3, pp. 165-174

Résumé

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.

Détail

DOI : 10.4064/ba8092-12-2016

Keywords: explicit formulas quadratic mean value dirichlet l functions associated set phi odd dirichlet characters mod known have obtain explicit upper bounds relative class numbers cyclotomic number fields here present generalization these results explicit formulas quadratic mean values dirichlet l functions associated subsets obtained application obtain explicit upper bounds relative class numbers imaginary subfields cyclotomic number fields