The Selberg Trace Formula for PSL (2,R) : Volume 1

During the last 10 years or so, mathematicians have become increasingly fascinated with the Selberg trace formula. These notes were written to help remedy this situation. Their main purpose is to provide a comprehensive development of the trace formula for PSL(2,R). Volume one deals exclusively with the case of compact quotient space. Although the trace formula can be developed much more generally, there are severe limitations on what is known for the higher-dimensional groups. Under these circumstances, it makes sense to try to understand the simplest cases first. The main chapter in volume one is chapter 2. In this chapter, we study the trace formula using the techniques of analytic number theory. By focusing on the Selberg zeta function, we can prove some deep results about the distribution of eigenvalues and pseudoprimes (lengths of closed geodesics) for a compact Riemann surface. This is the first time most of this material has appeared in print. Roughly speaking: 1/3 of it is fai