Algebra & Number Theory

Mean square in the prime geodesic theorem

Giacomo Cherubini and João Guerreiro

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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.

Article information

Algebra Number Theory, Volume 12, Number 3 (2018), 571-597.

Received: 23 May 2017
Revised: 26 October 2017
Accepted: 30 December 2017
First available in Project Euclid: 28 July 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11F72: Spectral theory; Selberg trace formula
Secondary: 11L05: Gauss and Kloosterman sums; generalizations 11M36: Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. Explicit formulas

prime geodesic theorem Selberg trace formula Kuznetsov trace formula Kloosterman sums


Cherubini, Giacomo; Guerreiro, João. Mean square in the prime geodesic theorem. Algebra Number Theory 12 (2018), no. 3, 571--597. doi:10.2140/ant.2018.12.571.

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