TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 352, Number 12, Pages 5563–5582 S 0002-9947(00)02502-2 Article electronically published on August 21, 2000 ASYMPTOTIC RELATIONS AMONG FOURIER COEFFICIENTS OF REAL-ANALYTIC EISENSTEIN SERIES ALVARO ALVAREZ-PARRILLA 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 ϕ ir with an exponential factor along a horocycle. We evaluate the integral in two ways by exploiting the automorphicity of ϕ 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 λ (λ = 1 4 + r 2 is the eigenvalue of ϕ ir ) for a cubic phase. As applications we find sets of asymptotic relations for divisor functions. 1. Preliminaries Let Γ ⊂ SL 2 (R) be a finitely generated non-cocompact Fuchsian group of the first kind. 1 For ease of presentation, we shall restrict ourselves to the case where Γ= SL 2 (Z), although most of the work carries through to the general case (the exception being in the Appendix, where one has to be careful about the choice of group so that it has a standard fundamental domain). Let ϕ s− 1 2 denote a Γ-automorphic eigenfunction of the hyperbolic Laplacian with eigenvalue λ = s(1 − s)= 1 4 + r 2 > 1 4 on the critical line Re(s)= 1 2 (we choose s such that s = 1 2 + ir) with a Fourier series development of the form ϕ s− 1 2 (z )= a 0+ y s + a 0− y 1−s +  m=0 a m y 1 2 K s− 1 2 (2π|m|y) e i2πmx , (1.1) where K q denotes the K-Bessel functions (also known as the MacDonald-Bessel functions), and z = x + iy ∈ H = {z ∈ C : y = Im(z ) > 0}. By saying that ϕ ir is Γ-automorphic for Γ ⊂ SL 2 (R), we mean that for γ = ( ab cd ) ∈ Γ ⊂ SL 2 (R), acting on the upper half plane H by linear fractional trans- formations z → az+b cz+d , we have ϕ ir (z )= ϕ ir (γz ). In the case that Γ = SL 2 (Z), we Received by the editors September 29, 1998 and, in revised form, November 24, 1998 and January 29, 1999. 1991 Mathematics Subject Classification. Primary 11F30; Secondary 11N37. Key words and phrases. Automorphic forms, Eisenstein series, microlocal analysis, divisor functions. Thanks to Scott Wolpert for suggesting the problem, many very insightful talks and helpful ideas, and for providing copies of his preprint. 1 In other words Γ is a subgroup of PSL 2 (R) acting discontinuously on H containing parabolic elements, and every point on the boundary ∂H is a limit of an orbit Γz for some z ∈ H. Poincar´ e showed that a subgroup of SL 2 (R) is discrete if and only if it acts discontinuously (when considered as a subgroup of PSL 2 (R)). c 2000 American Mathematical Society 5563 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use