On the Prime Geodesic Theorem for SL4

Abstract

In 1949, A. Selberg discovered a real variable (an elementary) proof of the prime number theorem. A number of authors have adapted Selberg’s method to achieve quite a good corresponding error term. The Riemann hypothesis has never been proved or disproved however. Any generalization of the prime number theorem to the more general situations is known in literature as a prime geodesic theorem. In this paper we derive yet another proof of the prime geodesic theorem for compact symmetric spaces formed as quotients of the Lie group SL4 (R). While the first known proof in this setting applies contour integration over square boundaries, our proof relies on an application of modified circular boundaries. Recently, A. Deitmar and M. Pavey applied such prime geodesic theorem to derive an asymptotic formula for class numbers of orders in totally complex quartic fields with no real quadratic subfields.