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 $|L(1,\chi ){|}^2$ 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 $|L(1,\chi ){|}^2$ is asymptotic to ${\pi }^2/6$, where $\chi$ runs over all odd primitive Dirichlet characters of prime conductors $p\equiv 1(\mathrm{mod}2d)$ which are trivial on a subgroup $H$ of odd order $d$ of the multiplicative group $(\mathbb{Z}/p\mathbb{Z}{)}^{\ast }$, 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 ${\chi }^{\prime }$ modulo $d_{0}p$ induced by the odd primitive characters $\chi$ modulo $p$. We obtain new estimates for Dedekind sums and deduce that the mean value of $|L(1,{\chi }^{\prime }){|}^2$ 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.

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