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Transactions of the American Mathematical Society

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Asymptotic relations among Fourier coefficients of real-analytic Eisenstein series


Author: Alvaro Alvarez-Parrilla
Journal: Trans. Amer. Math. Soc. 352 (2000), 5563-5582
MSC (1991): Primary 11F30; Secondary 11N37
Published electronically: August 21, 2000
MathSciNet review: 1675233
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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.


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Additional Information

Alvaro Alvarez-Parrilla
Affiliation: Department of Mathematics, University of Maryland at College Park, College Park, Maryland 20740
Address at time of publication: P.O. Box 435294, San Ysidro, California 92173
Email: aap@math.umd.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-00-02502-2
Keywords: Automorphic forms, Eisenstein series, microlocal analysis, divisor functions
Received by editor(s): September 29, 1998
Received by editor(s) in revised form: November 24, 1998, and January 29, 1999
Published electronically: August 21, 2000
Additional Notes: Thanks to Scott Wolpert for suggesting the problem, many very insightful talks and helpful ideas, and for providing copies of his preprint
Article copyright: © Copyright 2000 American Mathematical Society