Fourier coefficients of real analytic Eisenstein series at various cusps (II)

2020 ◽  
Author(s):  
Huake Liu ◽  
Tianqin Wang
Keyword(s):  
Eisenstein Series ◽  
Fourier Coefficients ◽  
Real Analytic

Dirichlet series of two variables, real analytic Jacobi–Eisenstein series of matrix index, and Katok–Sarnak type result

2018 ◽  
Vol 30 (6) ◽  
pp. 1437-1459 ◽  
Author(s):  
Yoshinori Mizuno
Keyword(s):  
Modular Form ◽  
Dirichlet Series ◽  
Eisenstein Series ◽  
Hyperbolic Space ◽  
Fourier Coefficients ◽  
Type Result ◽  
Higher Dimensional ◽  
Real Analytic

Abstract We study a real analytic Jacobi–Eisenstein series of matrix index and deduce several arithmetically interesting properties. In particular, we prove the followings: (a) Its Fourier coefficients are proportional to the average values of the Eisenstein series on higher-dimensional hyperbolic space. (b) The associated Dirichlet series of two variables coincides with those of Siegel, Shintani, Peter and Ueno. This makes it possible to investigate the Dirichlet series by means of techniques from modular form.

scholarly journals Eisenstein series and an asymptotic for the K-Bessel function

2021 ◽  
Author(s):  
Jimmy Tseng
Keyword(s):  
Asymptotic Expansion ◽  
Bessel Function ◽  
Eisenstein Series ◽  
Uniform Estimate ◽  
Positive Real ◽  
Dominant Term ◽  
Complex Order ◽  
Fixed Parameter ◽  
Real Argument ◽  
Real Analytic

AbstractWe produce an estimate for the K-Bessel function $$K_{r + i t}(y)$$ K r + i t ( y ) with positive, real argument y and of large complex order $$r+it$$ r + i t where r is bounded and $$t = y \sin \theta $$ t = y sin θ for a fixed parameter $$0\le \theta \le \pi /2$$ 0 ≤ θ ≤ π / 2 or $$t= y \cosh \mu $$ t = y cosh μ for a fixed parameter $$\mu >0$$ μ > 0 . In particular, we compute the dominant term of the asymptotic expansion of $$K_{r + i t}(y)$$ K r + i t ( y ) as $$y \rightarrow \infty $$ y → ∞ . When t and y are close (or equal), we also give a uniform estimate. As an application of these estimates, we give bounds on the weight-zero (real-analytic) Eisenstein series $$E_0^{(j)}(z, r+it)$$ E 0 ( j ) ( z , r + i t ) for each inequivalent cusp $$\kappa _j$$ κ j when $$1/2 \le r \le 3/2$$ 1 / 2 ≤ r ≤ 3 / 2 .

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.

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