Asymptotic relations among Fourier coefficients of real-analytic Eisenstein series

Abstract:

Following Wolpert, we find a set of asymptotic relations among the Fourier coefficients of real-analytic Eisenstein series. The relations are found by evaluating the integral of the product of an Eisenstein series $\varphi _{ir}$ with an exponential factor along a horocycle. We evaluate the integral in two ways by exploiting the automorphicity of $\varphi _{ir}$; the first of these evaluations immediately gives us one coefficient, while the other evaluation provides us with a sum of Fourier coefficients. The second evaluation of the integral is done using stationary phase asymptotics in the parameter $\lambda \ (\lambda =\frac {1}{4}+r^2$ is the eigenvalue of $\varphi _{ir}$) for a cubic phase. As applications we find sets of asymptotic relations for divisor functions.