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

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.