An asymptotic formula for the 2𝑘-th power mean value of |(𝐿′/𝐿)(1 + 𝑖𝑡₀, 𝜒)|

Abstract

Let 𝑞 be a positive integer ( ≥ 2), 𝜒 be a Dirichlet character modulo 𝑞, 𝐿(𝑠, 𝜒) be the attached Dirichlet 𝐿-function, and let 𝐿^′ (𝑠, 𝜒) denote its derivative with respect to the complex variable 𝑠. Let 𝑡₀ be any fixed real number. The main purpose of this paper is to give an asymptotic formula for the 2𝑘-th power mean value of |(𝐿^′/𝐿)(1 + 𝑖𝑡₀, 𝜒)| when 𝜒 runs over all Dirichlet characters modulo 𝑞 (except the principal character when 𝑡₀ = 0).