Stopple The aim of this investigation is to study certain L-functions on their critical line and to study the finer structure of the spectrum of the Laplace operator on certain completely integrable systems. The L-functions are complex analytic functions, given by power series in a domain of the complex plane, but their most interesting behavior is on their critical line, which belongs to the region of analytic continuation. The study of L-functions of Rankin-Selberg type that arise from cusp forms of finite area hyperbolic surfaces is related to the general Ramanujan conjecture. The methods to be used belong to the spectral theory of automorphic forms. The problems in the finer structure of the spectrum have recently attracted attention among physicists working on Quantum Chaos. The central problem is the distribution of the spacings between the eigenvalues. The hope is to bring tools from analytic number theory to the study of the pair correlation function of completely integrable systems. This research falls into the general mathematical field of Number Theory. Number theory has its historical roots in the study of the whole numbers, addressing such questions as those dealing with the divisibility of one whole number by another. It is among the oldest branches of mathematics and was pursued for many centuries for purely aesthetic reasons. However, within the last half century it has become an indispensable tool in diverse applications in areas such as data transmission and processing, and communication systems.
