4 - The Rankin–Selberg Method: A Survey

Abstract

This chapter presents a survey of the Rankin-Selberg method. The earliest applications of the Rankin-Selberg method were to the estimation of the Fourier coefficients of modular forms. The Rankin-Selberg method gives an improvement over the “trivial estimate”.. Hecke introduced operators on cusp forms for each p that play a role similar to the Laplacian. These operators are self-adjoint and mutually commutative, hence might be simultaneously diagonalized. Rankin and, independently at around the same time, Selberg introduced a new tool into the study of cusp forms, which is known as the Rankin-Selberg method. This gives the functional equation of a new kind of Euler product, and gives new estimates for the Fourier coefficients.