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

## TL;DR

This paper investigates the asymptotic behavior of generalized Dirichlet L-functions, extending classical results to a broader class and providing sharp asymptotic formulas through analytic methods.

## Contribution

It introduces and analyzes the mean value properties of generalized Dirichlet L-functions, deriving new sharp asymptotic formulas for their behavior.

## Key findings

- Derived sharp asymptotic formulas for mean values of generalized Dirichlet L-functions.
- Extended classical Dirichlet L-function results to a more general setting with parameter a.
- Provided analytic continuation and asymptotic analysis for these functions.

## 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.