On the mean value of the generalized Dirichlet L-functions with the weight of the Gauss Sums

2019/12/31 by Rong Ma, Ma, Rong, Yana Niu +1
Mathematics · #Analytic Number Theory Research #Advanced Mathematical Identities #Algebraic Geometry and Number Theory

paper · pdf · doi:10.48550/arxiv.1912.13153

Abstract

Let q≥3 be an integer, χ denote a Dirichlet character modulo q, for any real number a≥ 0, we define the generalized Dirichlet L-functions L(s,χ,a)=∑_n=1^∞(χ(n))/((n+a)^s), where s=σ+it with σ>1 and t both real. It can be extended to all s by analytic continuation. For any integer m, the famous Gauss sum G(m,χ) is defined as follows: G(m,χ)=∑_a=1^qχ(a)e((am)/(q)), where e(y)=e^2πiy. The main purpose of this paper is to use the analytic method to study the mean value properties of the generalized Dirichlet L-functions with the weight of the Gauss Sums, and obtain a sharp asymptotic formula.

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