Cancellation of Cusp Forms Coefficients over Beatty Sequences on GL(m)

2011 ◽  
Vol 54 (4) ◽  
pp. 757-762
Author(s):  
Qingfeng Sun
Keyword(s):  
Cusp Form ◽  
Fourier Coefficients ◽  
Cusp Forms ◽  
Maass Cusp Form ◽  
Beatty Sequences

AbstractLet A(n1, n2, … , nm–1) be the normalized Fourier coefficients of a Maass cusp form on GL(m). In this paper, we study the cancellation of A(n1, n2, … , nm–1) over Beatty sequences.

On sums of Fourier coefficients of Maass cusp forms

2017 ◽  
Vol 13 (05) ◽  
pp. 1233-1243 ◽  
Author(s):  
Yujiao Jiang ◽  
Guangshi Lü
Keyword(s):  
Fourier Coefficient ◽  
Cusp Form ◽  
Fourier Coefficients ◽  
Cusp Forms ◽  
Maass Cusp Form

Let [Formula: see text] be a Hecke–Maass cusp form, and [Formula: see text] be its [Formula: see text]th Fourier coefficient at the cusp infinity. In this paper, we are interested in the estimation on sums [Formula: see text] for [Formula: see text] We are able to improve previous results by introducing some inequalities concerning Fourier coefficients and other techniques.

scholarly journals Strong orthogonality between the Möbius function, additive characters and Fourier coefficients of cusp forms

2014 ◽  
Vol 150 (5) ◽  
pp. 763-797 ◽  
Author(s):  
Étienne Fouvry ◽  
Satadal Ganguly
Keyword(s):  
Real Number ◽  
Fourier Coefficient ◽  
Cusp Form ◽  
Fourier Coefficients ◽  
Möbius Function ◽  
Cusp Forms ◽  
Mobius Function ◽  
Maass Cusp Form ◽  
Strong Orthogonality

AbstractLet$\nu _{f}(n)$be the$n\mathrm{th}$normalized Fourier coefficient of a Hecke–Maass cusp form$f$for${\rm SL }(2,\mathbb{Z})$and let$\alpha $be a real number. We prove strong oscillations of the argument of$\nu _{f}(n)\mu (n) \exp (2\pi i n \alpha )$as$n$takes consecutive integral values.

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.

scholarly journals Fourier coefficients of Siegel cusp forms of degree two

1984 ◽  
Vol 93 ◽  
pp. 149-171 ◽  
Author(s):  
Yoshiyuki Kitaoka
Keyword(s):  
Cusp Form ◽  
Fourier Coefficients ◽  
Cusp Forms ◽  
Siegel Cusp Form

Our purpose is to prove the followingTheorem. Let k be an even integer ≥ 6. Letbe a Siegel cusp form of degree two, weight k. Then we have

scholarly journals On products of Fourier coefficients of cusp forms

2017 ◽  
Vol 29 (1) ◽  
Author(s):  
Eric Hofmann ◽  
Winfried Kohnen
Keyword(s):  
Cusp Form ◽  
Fourier Coefficients ◽  
Cusp Forms

AbstractThe purpose of this paper is to study products of Fourier coefficients of an elliptic cusp form,

scholarly journals Large sums of Hecke eigenvalues of holomorphic cusp forms

2019 ◽  
Vol 31 (2) ◽  
pp. 403-417
Author(s):  
Youness Lamzouri
Keyword(s):  
Cusp Form ◽  
Riemann Hypothesis ◽  
Modular Group ◽  
Fourier Coefficients ◽  
Automorphic Forms ◽  
Character Sums ◽  
Cusp Forms ◽  
Hecke Eigenvalues ◽  
Sign Change ◽  
Holomorphic Cusp Forms

AbstractLet f be a Hecke cusp form of weight k for the full modular group, and let {\{\lambda_{f}(n)\}_{n\geq 1}} be the sequence of its normalized Fourier coefficients. Motivated by the problem of the first sign change of {\lambda_{f}(n)}, we investigate the range of x (in terms of k) for which there are cancellations in the sum {S_{f}(x)=\sum_{n\leq x}\lambda_{f}(n)}. We first show that {S_{f}(x)=o(x\log x)} implies that {\lambda_{f}(n)<0} for some {n\leq x}. We also prove that {S_{f}(x)=o(x\log x)} in the range {\log x/\log\log k\to\infty} assuming the Riemann hypothesis for {L(s,f)}, and furthermore that this range is best possible unconditionally. More precisely, we establish the existence of many Hecke cusp forms f of large weight k, for which {S_{f}(x)\gg_{A}x\log x}, when {x=(\log k)^{A}}. Our results are {\mathrm{GL}_{2}} analogues of work of Granville and Soundararajan for character sums, and could also be generalized to other families of automorphic forms.

scholarly journals Petersson products of bases of spaces of cusp forms and estimates for Fourier coefficients

2018 ◽  
Vol 14 (08) ◽  
pp. 2277-2290 ◽  
Author(s):  
Rainer Schulze-Pillot ◽  
Abdullah Yenirce
Keyword(s):  
Cusp Form ◽  
Fourier Coefficients ◽  
Orthogonal Basis ◽  
Cusp Forms ◽  
Integral Weight ◽  
Space Of Cusp Forms

We prove a bound for the Fourier coefficients of a cusp form of integral weight which is not a newform by computing an explicit orthogonal basis for the space of cusp forms of given integral weight and level.

scholarly journals Estimates for fourier coefficients of siegel cusp forms of degree two, II

1992 ◽  
Vol 128 ◽  
pp. 171-176 ◽  
Author(s):  
Winfried Kohnen
Keyword(s):  
Cusp Form ◽  
Fourier Coefficients ◽  
Positive Definite ◽  
Cusp Forms ◽  
Integral Weight ◽  
Siegel Cusp Form

Let F be a Siegel cusp form of integral weight k on Γ2: = Sp2(Z) and denote by a(T) (T a positive definite symmetric half-integral (2,2)-matrix) its Fourier coefficients. In [2] Kitaoka proved that(1)(the result is actually stated only under the assumption that k is even). In our previous paper [3] it was shown that one can attain(2)

scholarly journals On LevelpSiegel Cusp Forms of Degree Two

2012 ◽  
Vol 2012 ◽  
pp. 1-9
Author(s):  
Hirotaka Kodama ◽  
Shoyu Nagaoka ◽  
Yoshitsugu Nakamura
Keyword(s):  
Cusp Form ◽  
Simple Formula ◽  
Fourier Coefficients ◽  
Cusp Forms ◽  
Siegel Cusp Form

We give a simple formula for the Fourier coefficients of some degree-two Siegel cusp form with levelp.

scholarly journals Analytic Twists of GL3 × GL2 Automorphic Forms

2021 ◽  
Author(s):  
Yongxiao Lin ◽  
Qingfeng Sun
Keyword(s):  
Cusp Form ◽  
Smooth Function ◽  
Fourier Coefficients ◽  
Automorphic Forms ◽  
Large Parameter ◽  
Hecke Eigenvalues ◽  
Maass Cusp Form

Abstract Let $\pi $ be a Hecke–Maass cusp form for $\textrm{SL}_3(\mathbb{Z})$ with normalized Hecke eigenvalues $\lambda _{\pi }(n,r)$. Let $f$ be a holomorphic or Maass cusp form for $\textrm{SL}_2(\mathbb{Z})$ with normalized Hecke eigenvalues $\lambda _f(n)$. In this paper, we are concerned with obtaining nontrivial estimates for the sum $$\begin{align*}& \sum_{r,n\geq 1}\lambda_{\pi}(n,r)\lambda_f(n)e\left(t\,\varphi(r^2n/N)\right)V\left(r^2n/N\right), \end{align*}$$where $e(x)=e^{2\pi ix}$, $V(x)\in \mathcal{C}_c^{\infty }(0,\infty )$, $t\geq 1$ is a large parameter and $\varphi (x)$ is some real-valued smooth function. As applications, we give an improved subconvexity bound for $\textrm{GL}_3\times \textrm{GL}_2$  $L$-functions in the $t$-aspect and under the Ramanujan--Petersson conjecture we derive the following bound for sums of $\textrm{GL}_3\times \textrm{GL}_2$ Fourier coefficients $$\begin{align*}& \sum_{r^2n\leq x}\lambda_{\pi}(r,n)\lambda_f(n)\ll_{\pi, f, \varepsilon} x^{5/7-1/364+\varepsilon} \end{align*}$$for any $\varepsilon&gt;0$, which breaks for the 1st time the barrier $O(x^{5/7+\varepsilon })$ in a work by Friedlander–Iwaniec.

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