Functional equations and characterizations of Riemann's function
type: Item
instance of: scholarly article
title: Functional equations and characterizations of Riemann's function
zbMATH Open document ID: 0901.39005
DOI: 10.1007/BF02109789
author: Yanyan Li
published in: Periodica Mathematica Hungarica
publication date: 1998-11-15T00:00:00Z
review text: The Riemann function is defined by \(r(x)= \sum^\infty_{\nu=1} {1\over \nu^2} \sin (\nu^2x)\) and the authors let \(R(x)= r(2\pi x)\) be a scaled Riemann function. It satisfies the functional equations \(R(x+1) =R(x)\), \(R(-x)= -R(x)\), and \(R({x \over 2}) +R({x+1 \over 2})= {1 \over 2} R(2x)\) for all real \(x\). It is shown that \(R\) satisfies \(\sum^{q-1}_{k=0} R({x+k \over q})= {1\over q} R(qx)\) for \(q\) a square free positive integer and any real \(x\) and that \(R({1\over 4}) =\pi^2/8\). Then in Theorem 2 the authors show that \(g=R\) is the only continuous solution of the functional equations \[ \sum^{p-1}_{k=0} g\left( {x+k \over p} \right)= {1\over p} g(px) \tag{*} \] where \(g:\mathbb{R} \to\mathbb{R}\) is continuous, 1-periodic, non-even, \(g({1\over 4}) =\pi^2/8\), \(p\) is any prime, and \(x\) is any real. The conditions of this theorem are somewhat weakened without losing the characterization of \(R\) in a Theorem 3, where \(g:\mathbb{R} \to\mathbb{R}\) is continuous on \((0,1)\), non-even, \(q\)-periodic for some prime \(q\), has the value \(\pi^2/8\) at 1/4, and satisfies (*) for all \(x\in \mathbb{R}\) and all primes \(p\).
Mathematics Subject Classification ID: 39B22
Mathematics Subject Classification ID: 39B62
Mathematics Subject Classification ID: 26A27
Mathematics Subject Classification ID: 26A30
zbMATH DE Number: 938669
zbMATH Keywords: functional equations
zbMATH Keywords: characterization
zbMATH Keywords: Riemann's continuous non-differentiable function
reviewed by: Konrad J. Heuvers
MaRDI profile type: Publication
cites work: The Differentiability of the Riemann Function at Certain Rational Multiples of &#960
cites work: More on the Differentiability of the Riemann Function
cites work: Pointwise analysis of Riemann's ``nondifferentiable'' function
cites work: Differentiability of Riemann's function
cites work: The differentiability of Fourier gap series and ``Riemann's example'' of a continuous, nondifferentiable function
cites work: Wo ist die Riemann-Funktion \(\sum_{n=1}^\infty \sin n^ 2 x / n^ 2\) nicht differenzierbar?
cites work: Riemann's example of a continuous, 'nondifferentiable' function
cites work: Riemann's example of a continuous ''nondifferentiable'' function continued
cites work: The Differentiability of Riemann's Function
description: scientific article; zbMATH DE number 938669