scholarly journals A new proof of the Voronoï summation formula

2011 ◽  
Vol 44 (22) ◽  
pp. 225302 ◽  
Author(s):  
Sebastian Egger né Endres ◽  
Frank Steiner
Keyword(s):  
Summation Formula ◽  
Voronoi Summation

Voronoi summation formula for Gaussian integers

2021 ◽  
Author(s):  
Debika Banerjee ◽  
Ehud Moshe Baruch ◽  
Daniel Bump
Keyword(s):  
Summation Formula ◽  
Gaussian Integers ◽  
Voronoi Summation

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.

SHIFTED CONVOLUTION SUM OF AND THE FOURIER COEFFICIENT OF HECKE–MAASS FORMS

2015 ◽  
Vol 92 (2) ◽  
pp. 195-204 ◽  
Author(s):  
HENGCAI TANG
Keyword(s):  
Fourier Coefficient ◽  
Circle Method ◽  
Summation Formula ◽  
Cusp Forms ◽  
Maass Forms ◽  
Average Value ◽  
Laplace Eigenvalue ◽  
Image Position ◽  
Voronoi Summation ◽  
Shifted Convolution Sum

Let $\{{\it\phi}_{j}(z):j\geq 1\}$ be an orthonormal basis of Hecke–Maass cusp forms with Laplace eigenvalue $1/4+t_{j}^{2}$. Let ${\it\lambda}_{j}(n)$ be the $n$th Fourier coefficient of ${\it\phi}_{j}$ and $d_{3}(n)$ the divisor function of order three. In this paper, by the circle method and the Voronoi summation formula, the average value of the shifted convolution sum for $d_{3}(n)$ and ${\it\lambda}_{j}(n)$ is considered, leading to the estimate $$\begin{eqnarray}\displaystyle \mathop{\sum }_{n\leq X}d_{3}(n){\it\lambda}_{j}(n-1)\ll X^{29/30+{\it\varepsilon}}, & & \displaystyle \nonumber\end{eqnarray}$$ where the implied constant depends only on $t_{j}$ and ${\it\varepsilon}$.

A SHIFTED CONVOLUTION SUM OF AND THE FOURIER COEFFICIENTS OF HECKE–MAASS FORMS II

2019 ◽  
Vol 101 (3) ◽  
pp. 401-414
Author(s):  
HENGCAI TANG
Keyword(s):  
Trace Formula ◽  
Fourier Coefficients ◽  
Summation Formula ◽  
Divisor Function ◽  
Maass Forms ◽  
Maass Form ◽  
Kuznetsov Trace Formula ◽  
Image Position ◽  
Voronoi Summation ◽  
Shifted Convolution Sum

Let $d_{3}(n)$ be the divisor function of order three. Let $g$ be a Hecke–Maass form for $\unicode[STIX]{x1D6E4}$ with $\unicode[STIX]{x1D6E5}g=(1/4+t^{2})g$. Suppose that $\unicode[STIX]{x1D706}_{g}(n)$ is the $n$th Hecke eigenvalue of $g$. Using the Voronoi summation formula for $\unicode[STIX]{x1D706}_{g}(n)$ and the Kuznetsov trace formula, we estimate a shifted convolution sum of $d_{3}(n)$ and $\unicode[STIX]{x1D706}_{g}(n)$ and show that $$\begin{eqnarray}\mathop{\sum }_{n\leq x}d_{3}(n)\unicode[STIX]{x1D706}_{g}(n-1)\ll _{t,\unicode[STIX]{x1D700}}x^{8/9+\unicode[STIX]{x1D700}}.\end{eqnarray}$$ This corrects and improves the result of the author [‘Shifted convolution sum of $d_{3}$ and the Fourier coefficients of Hecke–Maass forms’, Bull. Aust. Math. Soc.92 (2015), 195–204].

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 The Balanced Voronoi Formulas for $\textrm{GL}(n)$

2017 ◽  
Vol 2019 (11) ◽  
pp. 3473-3484 ◽  
Author(s):  
Stephen D Miller ◽  
Fan Zhou
Keyword(s):  
Summation Formula ◽  
Kloosterman Sums ◽  
Voronoi Summation

Abstract In this article, we show how the $\textrm{GL}(N)$ Voronoi summation formula of [13] can be rewritten to incorporate hyper-Kloosterman sums of various dimensions on both sides. This generalizes a formula for $\textrm{GL}(4)$ with ordinary Kloosterman sums on both sides that was used in [1] to prove nonvanishing of GL(4) $L$-functions by GL(2)-twists, and later by the second-named author in [16].

On an extension of the Hardy-Hilbert theorem

2005 ◽  
Vol 42 (1) ◽  
pp. 21-35 ◽  
Author(s):  
J. Weijian ◽  
G. Mingzhe ◽  
G. Xuemei
Keyword(s):  
Beta Function ◽  
Summation Formula ◽  
Hilbert’S Inequality ◽  
Two Parameters

A weighted Hardy-Hilbert’s inequality with the parameter λ of form \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\sum\limits_{m = 1}^\infty {\sum\limits_{n = 1}^\infty {\frac{{a_m b_n }}{{(m + n)^\lambda }}} < B^* (\lambda )\left( {\sum\limits_{n = 1}^\infty {n^{1 - \lambda } a_{a_n }^p } } \right)^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} \left( {\sum\limits_{n = 1}^\infty {n^{1 - \lambda } b_n^q } } \right)^q }$$ \end{document} is established by introducing two parameters s and λ, where \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\tfrac{1}{p} + \tfrac{1}{q} = 1,p \geqq q > 1,1 - \tfrac{q}{p} < \lambda \leqq 2,B^* (\lambda ) = B(\lambda - (1 - \tfrac{{2 - \lambda }}{p}),1 - \tfrac{{2 - \lambda }}{p})$$ \end{document} is the beta function. B *(λ) is proved to be best possible. A stronger form of this inequality is obtained by means of the Euler-Maclaurin summation formula.

Finite Difference Method on Non-Uniform Meshes for Time Fractional Diffusion Problem

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Haili Qiao ◽  
Aijie Cheng
Keyword(s):  
Theoretical Analysis ◽  
Fractional Derivative ◽  
Time Discretization ◽  
Summation Formula ◽  
Fractional Diffusion ◽  
Diffusion Problem ◽  
Initial Moment ◽  
Central Difference ◽  
Ideal Convergence ◽  
Derivative Term

AbstractIn this paper, we consider the time fractional diffusion equation with Caputo fractional derivative. Due to the singularity of the solution at the initial moment, it is difficult to achieve an ideal convergence rate when the time discretization is performed on uniform meshes. Therefore, in order to improve the convergence order, the Caputo time fractional derivative term is discretized by the {L2-1_{\sigma}} format on non-uniform meshes, with {\sigma=1-\frac{\alpha}{2}}, while the spatial derivative term is approximated by the classical central difference scheme on uniform meshes. According to the summation formula of positive integer k power, and considering {k=3,4,5}, we propose three non-uniform meshes for time discretization. Through theoretical analysis, different time convergence orders {O(N^{-\min\{k\alpha,2\}})} can be obtained, where N denotes the number of time splits. Finally, the theoretical analysis is verified by several numerical examples.

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