scholarly journals Kloosterman sums and Fourier coefficients of Eisenstein series

2018 ◽  
Vol 49 (2) ◽  
pp. 391-409 ◽  
Author(s):  
Eren Mehmet Kıral ◽  
Matthew P. Young
Keyword(s):  
Eisenstein Series ◽  
Fourier Coefficients ◽  
Kloosterman Sums

Non-random behavior in sums of modular symbols

2021 ◽  
pp. 1-25
Author(s):  
Alex Cowan
Keyword(s):  
Eisenstein Series ◽  
Spectral Decomposition ◽  
Fourier Coefficients ◽  
Dirichlet Character ◽  
Common Phenomenon ◽  
Modular Symbols ◽  
Dirichlet Characters ◽  
Random Behavior ◽  
The Common ◽  
Automorphic Functions

We give explicit expressions for the Fourier coefficients of Eisenstein series twisted by Dirichlet characters and modular symbols on [Formula: see text] in the case where [Formula: see text] is prime and equal to the conductor of the Dirichlet character. We obtain these expressions by computing the spectral decomposition of automorphic functions closely related to these Eisenstein series. As an application, we then evaluate certain sums of modular symbols in a way which parallels past work of Goldfeld, O’Sullivan, Petridis, and Risager. In one case we find less cancelation in this sum than would be predicted by the common phenomenon of “square root cancelation”, while in another case we find more cancelation.

scholarly journals Explicit formulas for local factors in the Euler products for Eisenstein series

1989 ◽  
Vol 113 ◽  
pp. 37-87 ◽  
Author(s):  
Paul Feit
Keyword(s):  
Dirichlet Series ◽  
Rational Function ◽  
Eisenstein Series ◽  
Fourier Coefficients ◽  
Simple Condition ◽  
Explicit Formulas ◽  
Local Factors ◽  
Euler Products ◽  
Infinite Sums

Our objective is to prove that certain Dirichlet series (in our variable q−s), which are defined by infinite sums, can be expressed as a product of an explicit rational function in q−s times an unknown polynomial M in q−s Moreover we show that M(q−s) is 1 if a simple condition is met. The Dirichlet series appear in the Euler products of Fourier coefficients for Eisenstein series. The series discussed below generalize the functions α0(N, q−s) used by Shimura in [12], and the theorem is an extension of Kitaoka’s result [5].

Fourier coefficients of generalized eisenstein series of degree two. I

1981 ◽  
Vol 65 (1) ◽  
pp. 115-135 ◽  
Author(s):  
Shin-ichiro Mizumoto
Keyword(s):  
Eisenstein Series ◽  
Fourier Coefficients

scholarly journals Fourier coefficients of an orthogonal Eisenstein series

1995 ◽  
Vol 168 (2) ◽  
pp. 345-381
Author(s):  
Jerry Shurman
Keyword(s):  
Eisenstein Series ◽  
Fourier Coefficients
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