Zeros on the critical line for Dirichlet series attached to certain cusp forms

1983 ◽  
Vol 264 (1) ◽  
pp. 21-37 ◽  
Author(s):  
James Lee hafner
Keyword(s):  
Dirichlet Series ◽  
Critical Line ◽  
Cusp Forms

scholarly journals On Dirichlet series attached to cusp forms and the Siegel-zero

1984 ◽  
Vol 25 (1) ◽  
pp. 107-119 ◽  
Author(s):  
F. Grupp
Keyword(s):  
Dirichlet Series ◽  
Cusp Form ◽  
Fourier Expansion ◽  
Dirichlet Character ◽  
Hecke Operators ◽  
Cusp Forms ◽  
Siegel Zero ◽  
Image Position

Let k be an even integer greater than or equal to 12 and f an nonzero cusp form of weight k on SL(2, Z). We assume, further, that f is an eigenfunction for all Hecke-Operators and has the Fourier expansionFor every Dirichlet character xmod Q we define

On dirichlet series related to certain cusp forms

1998 ◽  
Vol 38 (1) ◽  
pp. 64-76 ◽  
Author(s):  
A. Kačėnas ◽  
A. Laurinčikas
Keyword(s):  
Dirichlet Series ◽  
Cusp Forms

scholarly journals A SEMI-ADELIC KUZNETSOV FORMULA OVER NUMBER FIELDS

2013 ◽  
Vol 09 (07) ◽  
pp. 1649-1681 ◽  
Author(s):  
PÉTER MAGA
Keyword(s):  
Fourier Coefficients ◽  
Critical Line ◽  
Analytic Number Theory ◽  
Number Fields ◽  
Cusp Forms ◽  
Weight Functions ◽  
Weighted Sum ◽  
Admissible Weight ◽  
Bessel Transform ◽  
Kuznetsov Formula

In this paper, we prove a semi-adelic version of the Kuznetsov formula over number fields. This formula matches a weighted sum made of Fourier coefficients of cusp forms and Eisenstein series with a weighted sum of Kloosterman sums, the latter weight function is a kind of Bessel transform of the former one. We obtain a variant which is valid over all number fields. The admissible weight functions are important in applications, they depend on the archimedean parameters of the representations and show exponential decay. The automorphic vectors are not necessarily spherical in the archimedean aspect. Such formulas are proven to be useful in analytic number theory, e.g., in the estimate of L-functions on the critical line.

Rankin–Cohen brackets on Jacobi forms of several variables and special values of certain Dirichlet series

2019 ◽  
Vol 15 (05) ◽  
pp. 925-933
Author(s):  
Abhash Kumar Jha ◽  
Brundaban Sahu
Keyword(s):  
Dirichlet Series ◽  
Scalar Product ◽  
Fourier Coefficients ◽  
Differential Operators ◽  
Cusp Forms ◽  
Jacobi Forms ◽  
Linear Map ◽  
Special Values ◽  
Several Variables

We construct certain Jacobi cusp forms of several variables by computing the adjoint of linear map constructed using Rankin–Cohen-type differential operators with respect to the Petersson scalar product. We express the Fourier coefficients of the Jacobi cusp forms constructed, in terms of special values of the shifted convolution of Dirichlet series of Rankin–Selberg type. This is a generalization of an earlier work of the authors on Jacobi forms to the case of Jacobi forms of several variables.

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