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Euler products of Selberg zeta functions in the critical strip

Kaneko, Ikuya and Koyama, Shin-ya (2022) Euler products of Selberg zeta functions in the critical strip. Ramanujan Journal, 59 (2). pp. 437-458. ISSN 1382-4090. doi:10.1007/s11139-022-00550-y.

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We extend the region of convergence of Euler products of Selberg zeta functions beyond the boundary R(s) = 1 for congruence subgroups of SL₂(ℤ) if they are associated with nontrivial irreducible unitary representations. The region depends on the size of the lowest eigenvalue of the Laplacian and extends to R(s) ⩾ 3/4 under the Selberg eigenvalue conjecture. The method is based on the ideas of Ramanujan. For any unitary representation, we also establish a relation between the asymptotic behaviour of partial Euler products and the error term in the prime geodesic theorem.

Item Type:Article
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URLURL TypeDescription Paper
Kaneko, Ikuya0000-0003-4518-1805
Additional Information:© 2022 Springer Nature. Received 10 February 2021. Accepted 07 January 2022. Published 17 February 2022. Data Availability. Data sharing was not applicable to this article as no datasets were generated or analysed during the current study.
Subject Keywords:Selberg zeta functions; Euler products; Deep Riemann hypothesis; Generalised Riemann hypothesis; Prime geodesic theorem; Selberg eigenvalue conjecture
Issue or Number:2
Classification Code:Mathematics Subject Classification: 11M36
Record Number:CaltechAUTHORS:20220217-687250000
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Official Citation:Kaneko, I., Koyama, Sy. Euler products of Selberg zeta functions in the critical strip. Ramanujan J (2022).
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:113497
Deposited By: George Porter
Deposited On:17 Feb 2022 20:35
Last Modified:30 Nov 2022 18:31

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