Mean square values of $L$-functions over subgroups for non primitive characters, Dedekind sums and bounds on relative class numbers

TL;DR

This paper investigates the mean square values of certain L-functions over subgroups, providing new estimates for Dedekind sums, asymptotic formulas, and improved bounds on relative class numbers of cyclotomic subfields, with methods involving Mersenne primes.

Contribution

The authors extend previous results to non-primitive characters, derive explicit formulas, and improve bounds on class numbers using new estimates for Dedekind sums and properties of Mersenne primes.

Findings

Mean value of |L(1,χ')|^2 asymptotic to (π^2/6)∏_{q|d_0}(1 - 1/q^2)

New estimates for Dedekind sums are established

Improved bounds on the relative class number of cyclotomic subfields

Abstract

An explicit formula for the mean value of $\vert L(1,\chi)\vert^2$ is known, where $\chi$ runs over all odd primitive Dirichlet characters of prime conductors $p$. Bounds on the relative class number of the cyclotomic field ${\mathbb Q}(\zeta_p)$ follow. Lately the authors obtained that the mean value of $\vert L(1,\chi)\vert^2$ is asymptotic to $\pi^2/6$, where $\chi$ runs over all odd primitive Dirichlet characters of prime conductors $p\equiv 1\pmod{2d}$ which are trivial on a subgroup $H$ of odd order $d$ of the multiplicative group $({\mathbb Z}/p{\mathbb Z})^*$, provided that $d\ll\frac{\log p}{\log\log p}$. Bounds on the relative class number of the subfield of degree $\frac{p-1}{2d}$ of the cyclotomic field ${\mathbb Q}(\zeta_p)$ follow. Here, for a given integer $d_0>1$ we consider the same questions for the non-primitive odd Dirichlet characters $\chi'$ modulo $d_0p$ induced by the odd primitive characters $\chi$ modulo $p$. We obtain new estimates for Dedekind sums and deduce that the mean value of $\vert L(1,\chi')\vert^2$ is asymptotic to $\frac{\pi^2}{6}\prod_{q\mid d_0}\left (1-\frac{1}{q^2}\right )$, where $\chi$ runs over all odd primitive Dirichlet characters of prime conductors $p$ which are trivial on a subgroup $H$ of odd order $d\ll\frac{\log p}{\log\log p}$. As a consequence we improve the previous bounds on the relative class number of the subfield of degree $\frac{p-1}{2d}$ of the cyclotomic field ${\mathbb Q}(\zeta_p)$. Moreover, we give a method to obtain explicit formulas and use Mersenne primes to show that our restriction on $d$ is essentially sharp.