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

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 that we prove and compare with the original numerical results. 1 2 Introduction This paper presents computational and theoretical results concerning zeros of the Selberg zeta-function. The thesis [Fr] shows that it is possible to use the transfer operator to compute in a precise way zeros of the Selberg zeta-function, and carries out computations for Γ0(4) for a one-parameter family of characters. The results show how zeros of the Selberg zeta-function follow curves in the complex plane 1Classification: 11M36, 11F72, 37C30 Addresses • Mathematisch Instituut Universiteit Utrecht, Postbus 80010, NL-3508 TA Utrecht, Nederland email: [email protected] • Institut für Theoretische Physik, Technische Universität Clausthal, Leibnizstraße 10, 38678 Clausthal-Zellerfeld, Deutschland current address: Mathematics Institute, Zeeman Building, University of Warwick, Coventry CV4 7AL, United Kingdom email: [email protected] • Institut für Theoretische Physik, Technische Universität Clausthal, Leibnizstraße 10, 38678 Clausthal-Zellerfeld, Deutschland email: [email protected]