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Mean value of real Dirichlet characters using a double Dirichlet series

Part of: Zeta and $L$-functions: analytic theory Exponential sums and character sums

Published online by Cambridge University Press:  22 March 2023

Martin Čech
Martin Čech*
Affiliation:
Department of Mathematics and Statistics, Concordia University, 1455 Boulevard de Maisonneuve Ouest, Montreal, QC H3G 1M8, Canada Current address: Mathematics and Statistics, University of Turku, 201 00 Turku, Finland e-mail: martin.cech@utu.fi
*
e-mail: martin.cech@concordia.ca
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Abstract

We study the double character sum $\sum \limits _{\substack {m\leq X,\\m\mathrm {\ odd}}}\sum \limits _{\substack {n\leq Y,\\n\mathrm {\ odd}}}\left (\frac {m}{n}\right )$ and its smoothly weighted counterpart. 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.

Keywords

Analytic number theoryDirichlet charactersmultiple Dirichlet seriescharacter sums

MSC classification

Primary: 11M32: Multiple Dirichlet series and zeta functions and multizeta values
Secondary: 11L40: Estimates on character sums
Type
Article
Information
Canadian Mathematical Bulletin , Volume 66 , Issue 4 , December 2023 , pp. 1135 - 1151
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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