On the remainder in the weighted length spectrum for strictly hyperbolic Fuchsian groups

Abstract

In this paper, we consider the remainder in a weighted form of the length spectrum for compact Riemann surfaces of genus greater than or equal to two. Earlier, we conducted a similar research where we applied the Cauchy residue theorem over two different square boundaries, one of which intersected the corresponding critical line, and some, quite complex estimates for the logarithmic derivative of the associated zeta functions of Selberg and Ruelle. The main goal of this paper is to achieve the same length spectrum with the same remainder as in our previous study, but in a much simpler way.