Prime Geodesic Theorem for Compact Riemann Surfaces
As it is known, there have been a number of attempts to obtain precise estimates for the number of primes not exceeding x. A lot of them are related to the ones done by Chebyshev. Thus, a good deal is known about them and their limitations. The truth, or otherwise, of the Riemann hypothesis, however, has still not been established. In this paper we derive a prime geodesic theorem for a compact Riemann surface regarded as a quotient of the upper halfplane by a discontinuous group. We assume that the surface at case, considered as a compact Riemannian manifold, is equipped with classical Poincare metric. Our result follows from the standard theory of the zeta functions of Selberg and Ruelle. The closed geodesics in this setting are in one-to-one correspondence with the conjugacy classes of the corresponding group, so analysis conducted here is reminiscent of the relationship between the distribution of rational primes and Riemann zeta function. By analogy with the classical arithmetic case and the fact that the Riemann hypothesis is true in our setting, one would certainly expect to obtain an analogous error term in the prime geodesic theorem. Bearing in mind that the corresponding Selberg zeta funcion has much more zeros than the Riemann zeta, the latter is not satisfied however. Keywords—Compact Riemann surfaces, prime geodesic theorem, upper half-plane, zeta functions, Laplace operator.