scholarly journals Computing Maass cusp form on general hyperbolic torus

2017 ◽  
Author(s):  
Nor Syazana Shamsuddin ◽  
Hishamuddin Zainuddin ◽  
Chan Kar Tim
Keyword(s):  
Cusp Form ◽  
Maass Cusp Form

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.

scholarly journals Rapid computation of L-functions attached to Maass forms

2018 ◽  
Vol 14 (05) ◽  
pp. 1459-1485 ◽  
Author(s):  
Andrew R. Booker ◽  
Holger Then
Keyword(s):  
Cusp Form ◽  
Critical Line ◽  
Maass Forms ◽  
Maass Cusp Form

Let [Formula: see text] be a degree-[Formula: see text] [Formula: see text]-function associated to a Maass cusp form. We explore an algorithm that evaluates [Formula: see text] values of [Formula: see text] on the critical line in time [Formula: see text]. We use this algorithm to rigorously compute an abundance of consecutive zeros and investigate their distribution.

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.

Power moments of automorphic L-function attached to Maass forms

2016 ◽  
Vol 12 (02) ◽  
pp. 427-443
Author(s):  
Huafeng Liu ◽  
Shuai Li ◽  
Deyu Zhang
Keyword(s):  
Lower Bounds ◽  
Cusp Form ◽  
Asymptotic Formulas ◽  
Maass Forms ◽  
Maass Cusp Form ◽  
Power Moments

Let [Formula: see text] be a normalized Maass cusp form for [Formula: see text]. For [Formula: see text], we define [Formula: see text] [Formula: see text] as the supremum of all numbers [Formula: see text] such that [Formula: see text] where [Formula: see text] is the automorphic [Formula: see text]-function attached to [Formula: see text]. In this paper, we shall establish the lower bounds of [Formula: see text] for [Formula: see text] and obtain asymptotic formulas for the second, fourth and sixth powers of [Formula: see text].

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 $\varepsilon>0$, which breaks for the 1st time the barrier $O(x^{5/7+\varepsilon })$ in a work by Friedlander–Iwaniec.

scholarly journals Bounds for GL3L-functions in depth aspect

2019 ◽  
Vol 31 (2) ◽  
pp. 303-318 ◽  
Author(s):  
Qingfeng Sun ◽  
Rui Zhao
Keyword(s):  
Cusp Form ◽  
Prime Power ◽  
Dirichlet Character ◽  
Implied Constant ◽  
Maass Cusp Form ◽  
Primitive Dirichlet Character

AbstractLet f be a Hecke–Maass cusp form for {\mathrm{SL}_{3}(\mathbb{Z})} and χ a primitive Dirichlet character of prime power conductor {\mathfrak{q}=p^{\kappa}}, with p prime. We prove the subconvexity boundL\Big{(}\frac{1}{2},\pi\otimes\chi\Big{)}\ll_{p,\pi,\varepsilon}\mathfrak{q}^{% 3/4-3/40+\varepsilon}for any {\varepsilon>0}, where the dependence of the implied constant on p is explicit and polynomial.

scholarly journals On effective determination of Maass forms from central values of Rankin–Selberg L-function

2015 ◽  
Vol 27 (1) ◽  
Author(s):  
Ritabrata Munshi ◽  
Jyoti Sengupta
Keyword(s):  
Cusp Form ◽  
Maass Forms ◽  
Central Values ◽  
Maass Cusp Form

AbstractWe address the problem of identifying a Hecke–Maass cusp form

scholarly journals Voronoï summation via switching cusps

2021 ◽  
Vol 194 (4) ◽  
pp. 657-685
Author(s):  
Edgar Assing ◽  
Andrew Corbett
Keyword(s):  
Cusp Form ◽  
Fourier Expansion ◽  
Fourier Coefficients ◽  
Summation Formula ◽  
Local Theory ◽  
General Level ◽  
Whittaker Functions ◽  
Maass Cusp Form ◽  
Voronoi Summation

AbstractWe consider the Fourier expansion of a Hecke (resp. Hecke–Maaß) cusp form of general level N at the various cusps of $$\Gamma _{0}(N)\backslash \mathbb {H}$$ Γ 0 ( N ) \ H . We explain how to compute these coefficients via the local theory of p-adic Whittaker functions and establish a classical Voronoï summation formula allowing an arbitrary additive twist. Our discussion has applications to bounding sums of Fourier coefficients and understanding the (generalised) Atkin–Lehner relations.

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.

Non-vanishing of the first derivative of GL(3) ×GL(2) L-functions

2018 ◽  
Vol 14 (03) ◽  
pp. 847-869
Author(s):  
Guohua Chen ◽  
Xiaofei Yan
Keyword(s):  
Asymptotic Formula ◽  
Cusp Form ◽  
Orthogonal Basis ◽  
Cusp Forms ◽  
First Derivative ◽  
Maass Cusp Form ◽  
Center Point

Let [Formula: see text] be a fixed self-dual Hecke–Maass cusp form for [Formula: see text] and [Formula: see text] be an orthogonal basis of odd Hecke–Maass cusp forms for [Formula: see text]. We prove an asymptotic formula for the average of the first derivative of the Rankin–Selberg [Formula: see text]-function of [Formula: see text] and [Formula: see text] at the center point [Formula: see text]. This implies the non-vanishing results for the first derivative of these [Formula: see text]-functions at the center point [Formula: see text].

Sign in / Sign up

Export Citation Format

Share Document