The pointwise behavior of Riemann’s function

Frederik Broucke, Jasson Vindas

Type: Article

Publication Date: 2023-08-29

Citations: 2

DOI: https://doi.org/10.4171/jfg/137

Abstract

We present a new and simple method for the determination of the pointwise Hölder exponent of Riemann's function \sum_{n=1}^{\infty} n^{-2}\sin(\pi n^{2} x) at every point of the real line. In contrast to earlier approaches, where wavelet analysis and the theta modular group were needed for the analysis of irrational points, our method is direct and elementary, being only based on the following tools from number theory and complex analysis: the evaluation of quadratic Gauss sums, the Poisson summation formula, and Cauchy's theorem.

Locations

Journal of Fractal Geometry Mathematics of Fractals and Related Topics - View - PDF
arXiv (Cornell University) - View - PDF
Ghent University Academic Bibliography (Ghent University) - View - PDF

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Works Cited by This (25)

Action	Title	Year	Authors
+ PDF Chat	Introduction to Analytic Number Theory	1976	Tom M. Apostol
+	Multiplicative Number Theory	1980	H. Davenport
+ PDF Chat	Differentiability of Riemann's function	1981	Seiichi Itatsu
+	Wavelet methods for pointwise regularity and local oscillations of functions	1996	Stéphane Jaffard Yves Meyer
+	Multifractal behavior of polynomial Fourier series	2013	Fernando Chamizo Adrián Ubis
+ PDF Chat	Some problems of diophantine approximation: Part II. The trigonometrical series associated with the elliptic ϑ-functions	1914	G. H. Hardy J. E. Littlewood
+ PDF Chat	Wo ist die riemannsche funktion $$\sum\limits_{n = 1}^\infty {\frac{{\sin n^2 x}}{{n^2 }}}$$ nichtdifferenzierbar?	1980	Ernst Mohr
+	Pointwise analysis of Riemann's ?nondifferentiable? function	1991	M. Holschneider Ph. Tchamitchian
+	Some Fourier series with gaps	2007	Fernando Chamizo Adrián Ubis
+ PDF Chat	The spectrum of singularities of Riemann's function	1996	Stéphane Jaffard