Vol. 12, No. 3, 2018

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Mean square in the prime geodesic theorem

Giacomo Cherubini and João Guerreiro

Vol. 12 (2018), No. 3, 571–597
Abstract

We prove upper bounds for the mean square of the remainder in the prime geodesic theorem, for every cofinite Fuchsian group, which improve on average on the best known pointwise bounds. The proof relies on the Selberg trace formula. For the modular group we prove a refined upper bound by using the Kuznetsov trace formula.

Keywords
prime geodesic theorem, Selberg trace formula, Kuznetsov trace formula, Kloosterman sums
Mathematical Subject Classification 2010
Primary: 11F72
Secondary: 11L05, 11M36
Milestones
Received: 23 May 2017
Revised: 26 October 2017
Accepted: 30 December 2017
Published: 12 June 2018
Authors
Giacomo Cherubini
Max-Planck-Institut für Mathematik
Bonn
Germany
João Guerreiro
Max-Planck-Institut für Mathematik
Bonn
Germany