The pointwise behavior of Riemann's function

TL;DR

This paper introduces a straightforward method to determine the pointwise Hölder exponent of Riemann's function at all real points, bypassing complex wavelet and modular group techniques used previously.

Contribution

The paper presents a new, elementary approach based on number theory and complex analysis for analyzing the pointwise regularity of Riemann's function, especially at irrational points.

Findings

Method applies to all real points, including irrationals.

Simplifies previous complex analysis techniques.

Provides explicit pointwise Hölder exponents.

Abstract

We present a new and simple method for the determination of the pointwise H\"{o}lder 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.