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Prime geodesics and averages of the Zagier L-series

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

Published online by Cambridge University Press:  11 May 2021

OLGA BALKANOVA ,
DMITRY FROLENKOV  and
MORTEN S. RISAGER
OLGA BALKANOVA
Affiliation:
Steklov Mathematical Institute of Russian Academy of Sciences 8 Gubkina st., Moscow, 119991, Russia. e-mail: [email protected]
DMITRY FROLENKOV
Affiliation:
HSE University and Steklov Mathematical Institute of Russian Academy of Sciences 8 Gubkina st., Moscow, 119991, Russia. e-mail: [email protected]
MORTEN S. RISAGER
Affiliation:
Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen Ø, Denmark. e-mail: [email protected]
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Abstract

The Zagier L-series encode data of real quadratic fields. We study the average size of these L-series, and prove asymptotic expansions and omega results for the expansion. We then show how the error term in the asymptotic expansion can be used to obtain error terms in the prime geodesic theorem.

MSC classification

Primary: 11M36: Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. Explicit formulas
Secondary: 11F72: Spectral theory; Selberg trace formula 11M41: Other Dirichlet series and zeta functions
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 172 , Issue 3 , May 2022 , pp. 705 - 728
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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