论文标题
非原始字符的亚组的均值为$ l $ functions，dedekind sums and bounds in相对类号码
Mean square values of $L$-functions over subgroups for non primitive characters, Dedekind sums and bounds on relative class numbers
论文作者
论文摘要
$ \ vert l（1，χ）\ vert^2 $的平均值的明确公式是已知的，其中$χ$在所有奇数原始的dirichlet字符上运行$ p $。在Cyclotomic Field $ {\ Mathbb Q}（ζ_P）$的相对类中的界限。最近，作者获得了$ \ vert l（1，χ）\ vert^2 $的平均值是渐近的，对$π^2/6 $渐近，其中$χ$在所有奇特的原始dirichlet字符$ p \ oreiv 1 \ equiv 1 \ pmod pmod {2d} $上的奇数dirichlet tht $ $ $ $ $ $ $ h $ h $ h $ h $ h $ h $ z}/p {\ mathbb z}）^*$，提供$ d \ ll \ frac {\ log p} {\ log \ log \ log p} $。在$ \ frac {p-1} {2d} $的相对类数量上的界限，$ \ frac {p-1} {2d} $ of Cyclotomic field $ {\ Mathbb Q}（ζ_P）$ collage。在这里，对于给定的整数$ d_0> 1 $，我们考虑了非主要奇数dirichlet字符$χ'$ modulo $ d_0p $由奇数原始字符$χ$ modulo $ p $。我们获得了DEDEKIND总和的新估计，并推断出$ \ vert l（1，χ'）\ Vert^2 $的平均值是渐近的，对$ \ frac {π^2} {6} {6} \ prod_ {q \ mid d_0} Prime导体的dirichlet字符$ p $在子组$ h $上是奇数$ d \ d \ ll \ frac {\ log p} {\ log log \ log p} $。结果，我们改善了$ \ frac {p-1} {2d} $ cyclotomic field $ {\ mathbb q}（ζ_P）$的相对类数。此外，我们提供了一种获得明确公式的方法，并使用Mersenne Primes表明我们对$ D $的限制本质上是敏锐的。
An explicit formula for the mean value of $\vert L(1,χ)\vert^2$ is known, where $χ$ runs over all odd primitive Dirichlet characters of prime conductors $p$. Bounds on the relative class number of the cyclotomic field ${\mathbb Q}(ζ_p)$ follow. Lately the authors obtained that the mean value of $\vert L(1,χ)\vert^2$ is asymptotic to $π^2/6$, where $χ$ 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 ${\mathbb Q}(ζ_p)$ follow. Here, for a given integer $d_0>1$ we consider the same questions for the non-primitive odd Dirichlet characters $χ'$ modulo $d_0p$ induced by the odd primitive characters $χ$ modulo $p$. We obtain new estimates for Dedekind sums and deduce that the mean value of $\vert L(1,χ')\vert^2$ is asymptotic to $\frac{π^2}{6}\prod_{q\mid d_0}\left (1-\frac{1}{q^2}\right )$, where $χ$ 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 ${\mathbb Q}(ζ_p)$. Moreover, we give a method to obtain explicit formulas and use Mersenne primes to show that our restriction on $d$ is essentially sharp.
