Asymptotic relations among Fourier coefficients of real-analytic Eisenstein series
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- by Alvaro Alvarez-Parrilla PDF
- Trans. Amer. Math. Soc. 352 (2000), 5563-5582 Request permission
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.References
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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
- 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
- © Copyright 2000 American Mathematical Society
- 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
- MathSciNet review: 1675233