Mean value of real Dirichlet characters using a double Dirichlet series

: Čech Martin

: 2023

Canadian Mathematical Bulletin

: CANADIAN MATHEMATICAL BULLETIN-BULLETIN CANADIEN DE MATHEMATIQUES

: CAN MATH BULL

: 17

: 0008-4395

: 1496-4287

DOI: https://doi.org/10.4153/S000843952300022X

: https://doi.org/10.4153/S000843952300022X

We study the double character sum (m <= X, m odd n)Sigma (n <= Y n odd)Sigma (m/n ) and its smoothly weighted counter-part. An asymptotic formula with power saving error term was obtained by Conrey, Farmer, and Soundararajan by applying the Poisson summation formula. The result is interesting because the main term involves a non-smooth function. In this paper, we apply the inverse Mellin transform twice and study the resulting double integral that involves a double Dirichlet series. This method has two advantages-it leads to a better error term, and the surprising main term naturally arises from three residues of the double Dirichlet series.