Mean square in the prime geodesic theorem

Cherubini, Giacomo; Guerreiro, João

doi:10.2140/ant.2018.12.571

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Mean square in the prime geodesic theorem

Giacomo Cherubini and João Guerreiro

Vol. 12 (2018), No. 3, 571–597

DOI: 10.2140/ant.2018.12.571

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

Vol. 12, No. 3, 2018