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The second moment of symmetric square L-functions over Gaussian integers

Part of: Exponential sums and character sums Zeta and $L$-functions: analytic theory Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  11 January 2021

Olga Balkanova  and
Dmitry Frolenkov
Olga Balkanova
Affiliation:
Steklov Mathematical Institute of Russian Academy of Sciences, 8 Gubkina st., Moscow, 119991, Russia (balkanova@mi-ras.ru; frolenkov@mi-ras.ru)
Dmitry Frolenkov
Affiliation:
Steklov Mathematical Institute of Russian Academy of Sciences, 8 Gubkina st., Moscow, 119991, Russia (balkanova@mi-ras.ru; frolenkov@mi-ras.ru)
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Abstract

We prove a new upper bound on the second moment of Maass form symmetric square L-functions defined over Gaussian integers. Combining this estimate with the recent result of Balog–Biro–Cherubini–Laaksonen, we improve the error term in the prime geodesic theorem for the Picard manifold.

Keywords

Symmetric square L-functionsmomentsprime geodesic theorem

MSC classification

Primary: 11F12: Automorphic forms, one variable
Secondary: 11L05: Gauss and Kloosterman sums; generalizations 11M06: $zeta (s)$ and $L(s, chi)$
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 152 , Issue 1 , February 2022 , pp. 54 - 80
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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