The pointwise behavior of Riemann's function

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.

14 pages. An appendix on fractional integrals of modular forms was added

References in corpus (3)

• Riemann's non-differentiable function and the binormal curvature flow
• Geometric differentiability of Riemann's non-differentiable function
• On the Hausdorff dimension of Riemann's non-differentiable function

Cited by in corpus (1)

• Multifractality and intermittency in the limit evolution of polygonal vortex filaments