20 - Selberg's Lower Bound of the First Eigenvalue for Congruence Groups

Abstract

This chapter provides an overview of the Selberg's lower bound of the first Eigen value for congruence groups. The spectrum of Δ is related to the Kloosterman sums S(n,n;c) through the spectral representation of the zeta-function. Selberg has shown that Z(s) has meromorphic continuation to the whole complex s-plane. It is essential for the arguments to work with a finite sum of Kloosterman sums rather than with the zeta-function Z(s). The chapter illustrates an upper bound for Kloosterman sums.