On certain Dirichlet series built from the Fourier coefficients of modular functions

2004 ◽  
Vol 215 (1) ◽  
pp. 29-33
Author(s):  
Winfried Kohnen
Keyword(s):  
Dirichlet Series ◽  
Fourier Coefficients ◽  
Modular Functions

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 cusp forms

1986 ◽  
Vol 100 (1) ◽  
pp. 5-29 ◽  
Author(s):  
R. A. Rankin
Keyword(s):  
Modular Forms ◽  
Fourier Coefficients ◽  
Cusp Forms ◽  
Modular Functions ◽  
Field Of Study ◽  
Important Paper ◽  
Present Survey ◽  
Arithmetical Functions ◽  
Survey Article ◽  
Order Of Magnitude

The object of this survey article is to trace the influence on the theory of modular forms of the ideas contained in L. J. Mordell's important paper ‘On Mr Ramanujan's empirical expansions of modular functions’, which appeared in October 1917 in this Society's Proceedings [32]. The equally important paper [42] by S. Ramanujan, ‘On certain arithmetical functions’, referred to in Mordell's title, was published in May 1916 in the same Society's older journal, the Transactions, which was regrettably suppressed in 1928, 107 years after its foundation. Ramanujan's paper was concerned not only with multiplicative properties of Fourier coefficients of modular forms, but also with their order of magnitude. Since subsequent papers on the latter subject have also appeared in the Proceedings, it seems appropriate to include further developments in this field of study in the present survey.

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.

Some remarks on the Fourier coefficients of cusp forms

2020 ◽  
Vol 16 (09) ◽  
pp. 1935-1943
Author(s):  
Balesh Kumar ◽  
Jay Mehta ◽  
G. K. Viswanadham
Keyword(s):  
Fourier Coefficients ◽  
Cusp Forms ◽  
Modular Functions ◽  
Sign Changes ◽  
Half Integral Weight ◽  
Integral Weight

In this paper, we consider the angular changes of Fourier coefficients of half integral weight cusp forms and sign changes of [Formula: see text]-exponents of generalized modular functions.

Modular Functions and Dirichlet Series in Number Theory

1991 ◽  
Vol 75 (472) ◽  
pp. 249
Author(s):  
Steve Abbott ◽  
T. M. Apostol
Keyword(s):  
Number Theory ◽  
Dirichlet Series ◽  
Modular Functions

ARITHMETIC PROPERTIES OF TRACES OF SINGULAR MODULI ON CONGRUENCE SUBGROUPS

2010 ◽  
Vol 06 (08) ◽  
pp. 1755-1768 ◽  
Author(s):  
SOON-YI KANG ◽  
CHANG HEON KIM
Keyword(s):  
Modular Form ◽  
Modular Group ◽  
Fourier Coefficients ◽  
Singular Values ◽  
Modular Functions ◽  
Congruence Subgroups ◽  
Arithmetic Properties ◽  
Holomorphic Modular Form ◽  
Singular Moduli ◽  
Arbitrary Level

After Zagier proved that the traces of singular moduli are Fourier coefficients of a weakly holomorphic modular form, various arithmetic properties of the traces of singular values of modular functions mostly on the full modular group have been found. The purpose of this paper is to generalize the results for modular functions on congruence subgroups with arbitrary level.

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