Perturbation of Zeros of the Selberg Zeta Function for Γ₀(4)

Abstract

We study the asymptotic behavior of zeros of the Selberg zeta function for the congruence subgroup Γ₀(4) as a function of a one-parameter family of characters tending to the trivial character. The motivation for the study comes from observations based on numerical computations. Some of the observed phenomena lead to precise theorems, which we prove and compare with the original numerical results.

Supplementary material is available for this article. See the publisher's online edition of Experimental Mathematics for the following free supplementary resources: animations of the zeros of β↦Z(α, β) as a function of α.

2000 AMS Subject Classification:

• 11M36
• 11F72
• 37C30

Keywords:

• Selberg zeta functions and determinants
• spectral theory
• Selberg trace formula
• zeta functions
• transfer operators

ACKNOWLEDGMENTS

We thank the referee for his or her remarks and suggestions. The first author thanks D. Mayer for several invitations to visit Clausthal and is grateful to the Volkswagenstiftung for its financial support.