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

DOI:
https://doi.org/10.1090/S0002-9947-00-02502-2

Published electronically:
August 21, 2000

MathSciNet review:
1675233

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Abstract | References | Similar Articles | Additional Information

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 with an exponential factor along a horocycle. We evaluate the integral in two ways by exploiting the automorphicity of ; 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 is the eigenvalue of ) 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:
https://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