Jasson Vindas (Ghent University) : The pointwise behavior of Riemann's function

Orateur : Jasson Vindas

Affiliation : Ghent University

Date : 27 Janvier 2023

Heure : 14h

Lieu : Salle Kampé de Fériet, Bâtiment M2

Titre : The pointwise behavior of Riemann's function

Résumé : 

According to Weierstrass, Riemann would have proposed the function
\[
\sum_{n=1}^{\infty} \frac{\sin(\pi n^2 x)}{n^{2}}
\]
as an example of a nowhere differentiable function. The regularity analysis of this function has attracted much attention for more than a century, and culminated with the determination of its exact pointwise H\"{o}lder exponents at irrational numbers, which has also revealed that Riemann's function furnishes an example of true multifractal behavior.

In this talk we will present a new and simple method for the determination of the pointwise H\"{o}lder exponent of Riemann's function
at every point of the real line. In contrast to other 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.

The talk is based on collaborative work with Frederik Broucke.