Modular Forms and Dirichlet Series

Abstract

It was Ramanujan who in a fundamental paper of 1916 introduced his τ-function as the Fourier coefficient of a modular form and then attached a Dirichlet series to it. He established an analytic continuation of the series and a functional equation for it. He then made his famous conjectures about the multiplicativity of these coefficients and their size. The multiplicativity conjecture would allow him to write his Dirichlet series as an Euler product thereby establishing an analogy with classical zeta and L-functions. Subsequently Mordell proved that τ(n) is a multiplicative function but it was left to Hecke to develop a more elaborate theory and establish the existence of an infinite family of such examples. Ramanujan’s conjecture on estimating the size of τ(n) however defied immediate attack. The fundamental method of Rankin and Selberg did allow one to get good estimates for them but they were not optimal. The final resolution of Ramanujan’s conjecture came from algebraic geometry when it was shown to be a consequence of Deligne’s proof of the celebrated Weil conjectures. In this chapter, we will give a brief introduction to the fundamental concepts and study the oscillations of the Fourier coefficients from the standpoint of the non-vanishing of various L-functions.