Mellin transforms and Fourier-Ramanujan expansions
type: Item
instance of: scholarly article
title: Mellin transforms and Fourier-Ramanujan expansions
zbMATH Open document ID: 0595.10037
DOI: 10.1007/BF01160472
author: Dieter Klusch
published in: Mathematische Zeitschrift
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173802
review text: For Re \(\alpha\geq 0\) let \(K_{\alpha}\) be the class of all arithmetic functions \(f_{\alpha}(n)=\sum_{d | n}d^{1-\alpha} w(d),\) where \(w=M\{F\}\) is an inverse Mellin transform with \(F\in L(-\infty,+\infty)\). In this paper it is proved that each \(f_{\alpha}\in K_{\alpha}\) has an absolutely convergent Fourier-Ramanujan expansion \(f_{\alpha}(n)=\sum_{q\geq 1}a_ q(f_{\alpha}) c_ q(n),\) where \(c_ q(n)\) is Ramanujan's trigonometric sum and the Fourier coefficients \(a_ q(f_{\alpha})\) are integrals of Mellin type represented conversely by a modified Ramanujan series. The method of proof is based on the theory of Fourier transforms and on Ramanujan's classical Fourier expansion \(\sigma_{1-s}(n)=\zeta (s)\sum_{q\geq 1}q^{-s} c_ q(n)\) (Re s\(>1\), \(\sigma_ s(n)=\sum_{d | n}d^ s,\) \(\zeta\) (s) the Riemann zeta-function). In the applications some examples from the classes \(K_ 0\) and \(K_ 1\) are investigated including e.g. the logarithms of Jacobi's elliptic theta-functions. It here turns out that Ramanujan's outstanding identities \(0=\sum_{q\geq 1}q^{-1} c_ q(n)\) and \(-\sigma_ 0(n)=\sum_{q\geq 1}q^{-1} \log q\quad c_ q(n),\) which do not fit into the popular theory of Ramanujan expansions, are useful tools in the author's scheme of Fourier analysis for the class \(K_{\alpha}\).
Mathematics Subject Classification ID: 11N99
Mathematics Subject Classification ID: 11K55
Mathematics Subject Classification ID: 11A25
Mathematics Subject Classification ID: 11K65
zbMATH DE Number: 3957233
zbMATH Keywords: arithmetic functions
zbMATH Keywords: Fourier-Ramanujan expansion
zbMATH Keywords: Ramanujan trigonometric sum
zbMATH Keywords: Fourier coefficients
zbMATH Keywords: integrals of Mellin type
zbMATH Keywords: logarithms of Jacobi elliptic theta-functions
MaRDI profile type: Publication
cites work: Q4097001
cites work: Analytic Eisenstein series, theta-functions, and series relations in the spirit of Ramanujan.
cites work: Q5536163
cites work: On Ramanujan expansions of certain arithmetical functions
cites work: Essai sur l'application de la théorie des fonctions presque périodiques à l'arithmétique
cites work: Über die punktweise Konvergenz von Ramanujan-Entwicklungen zahlentheoretischer Funktionen
cites work: Auswertung einiger spezieller unendlicher Reihen aus dem Bereich der elliptischen Funktionen
cites work: On the approximation of analytic functions in a strip
cites work: Q4083499
cites work: Q3248054
cites work: Über die Ramanujan-Entwicklung multiplikativer Funktionen
cites work: Über die Entwicklung multiplikativer Funktionen nach Ramanujan-Summen
cites work: Q5834942
OpenAlex ID: W1978044558
description: scientific article; zbMATH DE number 3957233