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
Full-text PDF Free Access
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