Series Abstract

The original Ramanujan Conjectures concern the size of the Fourier coefficients of holomorphic modular forms.This can be interpreted either as a question about the spectra of Hecke Operators or as one about the size of a period of the modular form. For automorphic forms on general groups these two points points of view don~Rtcoincide and understanding each separately and establishing approximations towards the conjectured true bounds, is a central problem .We will discuss some of what is expected and what known, as well as the basic techniques that have been applied. The known approximations have many applications in both number theory and problems of mathematical physics associated with eigenfunctions on locally symmetric spaces. We discuss some recent applications to problems of sieving for primes and square free numbers, on homogeneous varieties. We will try to keep the lectures accessible to a general mathematical audience.