The Voronoi identity via the Laplace transform
type: Item
instance of: scholarly article
title: The Voronoi identity via the Laplace transform
zbMATH Open document ID: 0917.11046
DOI: 10.1023/A:1009753723245
published in: The Ramanujan Journal
publication date: 1998-11-08T00:00:00Z
review text: The famous Voronoi identity is a series representation involving Bessel functions for the error term \(\Delta (x)\) in the asymptotic formula for the sum function of the divisor function \(d(n)\). Known proofs for this result are fairly complicated. The idea of the author is to show first that the Laplace transform of \(\Delta (x)\) and that of its asserted series expression coincide, and to deduce then that the original functions coincide as well. A virtue of this argument is its applicability to sum functions of other arithmetic functions such as \(r(n)\) (the number of representations of \(n\) as the sum of two squares) or the Fourier coefficients of a cusp form.
reviewed by: Matti Jutila
Mathematics Subject Classification ID: 11N37
Mathematics Subject Classification ID: 11F30
Mathematics Subject Classification ID: 33C10
Mathematics Subject Classification ID: 44A10
zbMATH DE Number: 1203256
zbMATH Keywords: Voronoi identity
zbMATH Keywords: number of divisors
zbMATH Keywords: Laplace transform
zbMATH Keywords: Fourier coefficients of cusp forms
zbMATH Keywords: Bessel functions
author: Aleksandar Ivić
MaRDI profile type: Publication
full work available at URL: a:1009753723245
OpenAlex ID: W4252620675
Recommended article: Voronoi-type identity for a class of arithmetical functions via the Laplace transform
Recommended article: The Voronoi formula and double Dirichlet series
Recommended article: The Poisson Voronoi Tessellation I. A Basic Identity
Recommended article: Voronoi–Nasim summation formulas and index transforms
Recommended article: A relationship between Gale transforms and Voronoi diagrams
Recommended article: Q4203457
Recommended article: Application of an idea of Voronoĭ to lattice zeta functions
Recommended article: Q4013888
Recommended article: Q3748349
Recommended article: Another look at Voronovskaja type formulas
description: scientific article; zbMATH DE number 1203256