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arXiv · 1902.03846

On asymptotic properties of the generalized Dirichlet $L$-functions

Rong Ma·Yana Niu·Yulong Zhang
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Abstract

Let $q\ge3$ be an integer, $\chi$ denote a Dirichlet character modulo $q$, for any real number $a\ge 0$, we define the generalized Dirichlet $L$-functions $$ L(s,\chi,a)=\sum_{n=1}^{\infty}\frac{\chi(n)}{(n+a)^s}, $$ where $s=\sigma+it$ with $\sigma>1$ and $t$ both real. It can be extended to all $s$ by analytic continuation. In this paper, we study the mean value properties of the generalized Dirichlet $L$-functions, and obtain several sharp asymptotic formulae by using analytic method.

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Rong Ma, Yana Niu, Yulong Zhang. 2019-02-11. On asymptotic properties of the generalized Dirichlet $L$-functions. https://arxiv.org/abs/1902.03846

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