THE MEAN SQUARE OF THE LOGARITHMIC DERIVATIVE OF THE SELBERG ZETA FUNCTION FOR COCOMPACT DISCRETE SUBGROUPS

Abstract

In this paper, we study the mean square of the logarithmic derivative of the Selberg zeta function for cocompact discrete subgroups. Our results are analogues of the results on the mean square of the logarithmic derivative of the Riemann zeta function by Goldston, Gonek,and Montgomery (J. Reine Angew. Math. 537 (2001), 105-126). We obtain an asymptotic formula for the mean square of the logarithmic derivative of the Selberg zeta function,including a term on the pair correlation of the zeros of the Selberg zeta function. In addition,we introduce an integral related to the prime geodesic theorem in short intervals and prove that the integral is bounded by the mean square of the logarithmic derivative of the Selberg zeta function. The upper bound for the integral is improved in the case of the Selberg zeta function for arithmetic cocompact groups by proving an asymptotic formula for the mean square near the left side of the vertical line whose real part is one.