On the Hausdorff dimension of Riemann's non-differentiable function

TL;DR

This paper investigates the Hausdorff dimension of Riemann's non-differentiable function, linking its geometric properties to physical phenomena and providing upper estimates with multifractal extensions.

Contribution

It offers the first upper bound estimate of the Hausdorff dimension of Riemann's function and adapts these results to the multifractal setting, connecting geometric and physical aspects.

Findings

Upper estimate of Hausdorff dimension provided

Connections established with Talbot effect and Gauss sums

Recalculation of asymptotic behavior around rationals

Abstract

Recent findings show that the classical Riemann's non-differentiable function has a physical and geometric nature as the irregular trajectory of a polygonal vortex filament driven by the binormal flow. In this article, we give an upper estimate of its Hausdorff dimension. We also adapt this result to the multifractal setting. To prove these results, we recalculate the asymptotic behavior of Riemann's function around rationals from a novel perspective, underlining its connections with the Talbot effect and Gauss sums, with the hope that it is useful to give a lower bound of its dimension and to answer further geometric questions.