On Zeros of the Derivative of the Modified Selberg Zeta Function for the Modular Group

Asymptotic formulas for the number of zeros of the derivative of Selberg zeta functions were investigated in the celebrated paper due to Wenzhi Luo as an attempt to develop the study on the multiplicity of eigenvalues of hyperbolic Laplacians. The present paper is a succeeding study of the work, that is, it will deal with corresponding formulas for the modular group. In this case the Selberg zeta function has zeros which come from non-trivial zeros of the Riemann zeta function. But they are not our objects of this study. Therefore to remove these zeros, we shall define "the modified Selberg zeta function" and give the asymptotic formula for the vertical number of zeros of the derivative of the function. Moreover the paper deals with horizontal distribution of zeros.