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.

scholarly journals Quadratic congruences for Cohen–Eisenstein series

1999 ◽  
Vol 41 (1) ◽  
pp. 141-144
Author(s):  
P. GUERZHOY
Keyword(s):  
Number Theory ◽  
Eisenstein Series ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Analytic Number Theory ◽  
Cusp Forms ◽  
Special Values ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Published Paper

The notion of quadratic congruences was introduced in the recently published paper [A. Balog, H. Darmon and K. Ono, Congruences for Fourier coefficients of half-integral weight modular forms and special values of L-functions, in Analytic Number Theory, Vol. 1, Progr. Math.138 (Birkhäuser, Boston, 1996), 105–128.]. In this note we present different, somewhat more conceptual proofs of several results from that paper. Our method allows us to refine the notion and to generalize the results quoted. Here we deal only with the quadratic congruences for Cohen–Eisenstein series. Similar phenomena exist for cusp forms of half-integral weight as well; however, as one would expect, in the case of Eisenstein series the argument is much simpler. In particular, we do not make use of techniques other than p-adic Mazur measure, whereas the consideration of cusp forms of half-integral weight involves a much more sophisticated construction. Moreover, in the case of Cohen–Eisenstein series we are able to obtain a full and exhaustive result. For these reasons we present the argument here.

Titchmarsh's theorem of Hankel transform

2019 ◽  
pp. 11-24
Author(s):  
Mohamed-Ahmed Boudref
Keyword(s):  
Number Theory ◽  
Data Analysis ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Lipschitz Condition ◽  
Analytic Number Theory ◽  
Hankel Transform ◽  
Partial Differential ◽  
Analytic Number ◽  
Bessel Transform

Hankel transform (or Fourier-Bessel transform) is a fundamental tool in many areas of mathematics and engineering, including analysis, partial differential equations, probability, analytic number theory, data analysis, etc. In this article, we prove an analog of Titchmarsh's theorem for the Hankel transform of functions satisfying the Hankel-Lipschitz condition.

scholarly journals On triple correlations of Fourier coefficients of cusp forms

2018 ◽  
Vol 183 ◽  
pp. 485-492 ◽  
Author(s):  
Guangshi Lü ◽  
Ping Xi
Keyword(s):  
Fourier Coefficients ◽  
Cusp Forms

On linear relations for special L-values over certain totally real number fields

2021 ◽  
pp. 1-18
Author(s):  
Seiji Kuga
Keyword(s):  
Real Number ◽  
Fourier Coefficients ◽  
Number Fields ◽  
Arithmetic Functions ◽  
Hilbert Modular Form ◽  
Linear Relations ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Hilbert Modular ◽  
Totally Real

In this paper, we give linear relations between the Fourier coefficients of a special Hilbert modular form of half integral weight and some arithmetic functions. As a result, we have linear relations for the special [Formula: see text]-values over certain totally real number fields.

Exponential sums twisted by Fourier coefficients of automorphic cusp forms for SL(2, ℤ)

2014 ◽  
Vol 11 (01) ◽  
pp. 39-49 ◽  
Author(s):  
Bin Wei
Keyword(s):  
Asymptotic Expansion ◽  
Cusp Form ◽  
Rational Number ◽  
Fourier Coefficients ◽  
Bessel Functions ◽  
Exponential Sums ◽  
Summation Formula ◽  
Cusp Forms ◽  
Real Numbers ◽  
Voronoi Summation

Let f be a holomorphic cusp form of weight k for SL(2, ℤ) with Fourier coefficients λf(n). We study the sum ∑n>0λf(n)ϕ(n/X)e(αn), where [Formula: see text]. It is proved that the sum is rapidly decaying for α close to a rational number a/q where q2 < X1-ε. The main techniques used in this paper include Dirichlet's rational approximation of real numbers, a Voronoi summation formula for SL(2, ℤ) and the asymptotic expansion for Bessel functions.

Higher Moments of Fourier Coefficients of Cusp Forms

2015 ◽  
Vol 58 (3) ◽  
pp. 548-560
Author(s):  
Guangshi Lü ◽  
Ayyadurai Sankaranarayanan
Keyword(s):  
Modular Group ◽  
Fourier Coefficients ◽  
Arithmetic Function ◽  
Arithmetic Functions ◽  
Cusp Forms ◽  
Higher Moments ◽  
Integral Weight ◽  
Holomorphic Cusp Forms

AbstractLet Sk(Γ) be the space of holomorphic cusp forms of even integral weight k for the full modular group SL(z, ℤ). Let be the n-th normalized Fourier coefficients of three distinct holomorphic primitive cusp forms , and h(z) ∊ Sk3 (Γ), respectively. In this paper we study the cancellations of sums related to arithmetic functions, such as twisted by the arithmetic function λf(n).

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