Refereed journal article or data article (A1)
Mean value of real Dirichlet characters using a double Dirichlet series
List of Authors: Čech Martin
Publisher: CAMBRIDGE UNIV PRESS
Publication year: 2023
Journal: Canadian Mathematical Bulletin
Journal name in source: CANADIAN MATHEMATICAL BULLETIN-BULLETIN CANADIEN DE MATHEMATIQUES
Journal acronym: CAN MATH BULL
Number of pages: 17
ISSN: 0008-4395
eISSN: 1496-4287
DOI: http://dx.doi.org/10.4153/S000843952300022X
URL: https://doi.org/10.4153/S000843952300022X
Abstract
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.
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.
UTU Research Portal