13 - Integral Representations of Eisenstein series and L-functions

Abstract

This chapter illustrates the integral representations of Eisenstein series and the L-functions. The idea of obtaining integral formulas by integrating against some sort of Eisenstein series (or restriction of Eisenstein series) is not new. The “Rankin-Selberg method” is one idea that has interesting analytic and arithmetic consequences. A different use of Eisenstein series has been considered, obtaining not only integral representations of some L-functions, but also the integral representations of complicated Eisenstein series in terms of simpler ones. For the case of holomorphic cuspforms, the local integrals at infinite primes are expressible in terms of gamma functions. It seems to be an open problem to determine the precise nature of these Archimedean integrals in general.