A Certain Mean Square Value Involving Dirichlet L-Functions

Wenpeng Zhang,Di Han

doi:10.3390/math8060948

Abstract

The main purpose of this article is using the elementary methods, the properties of Dirichlet L-functions to study the computational problem of a certain mean square value involving Dirichlet L-functions at positive integer points, and give some exact calculating formulae. As some applications, we obtain some interesting identities and inequalities involving character sums and trigonometric sums.

Highlights

Let q ≥ 3 be an integer, χ denotes a Dirichlet character mod q
This paper, as a note of [7,17], we will use the elementary methods and the properties of Dirichlet L-functions to study the computational problem of one kind of special mean square value of Dirichlet L-functions, and give a new and exact calculating formula for it
Σk,u for all positive integers 0 ≤ u ≤ k − 1. It is clear this formula implies that Lemma 2 is correct for positive k + 1

Summary

Introduction

Let q ≥ 3 be an integer, χ denotes a Dirichlet character mod q. X. Lin [7] proved a general mean square value formula for Dirichlet L-functions L(n, χ). This paper, as a note of [7,17], we will use the elementary methods and the properties of Dirichlet L-functions to study the computational problem of one kind of special mean square value of Dirichlet L-functions, and give a new and exact calculating formula for it. Let q > 2 be an integer and χ denote a Dirichlet character mod q. For integers k = 1 and 2, from these theorems we may immediately deduce the following results: Corollary 1 ([6,7]).

Several Lemmas

Proofs of the Theorems

Conclusions