The a-values of the Selberg zeta-function

Abstract

We study the roots (a-values) of Z(s) = a, where Z(s) is the Selberg zeta-function attached to a compact Riemann surface. We obtain an asymptotic formula for the number of nontrivial a-values. If a ≠ 0, we show that the analogue of the Riemann hypothesis fails for nontrivial a-values; on other hand, almost all nontrivial a-values are arbitrarily close to the critical line. We also compare distributions of a-values for the Selberg and the Riemann zetafunctions.