Geometric differentiability of Riemann's non-differentiable function

TL;DR

This paper explores the geometric regularity of Riemann's non-differentiable function, showing that its trajectory in the complex plane lacks a tangent at any point, linking it to vortex filament evolution.

Contribution

It provides a geometric interpretation of Riemann's function and demonstrates its trajectory has no tangent anywhere, connecting it to vortex filament dynamics.

Findings

The trajectory of Riemann's function has no tangent at any point.

The geometric irregularity is analyzed using concepts of tangent vectors.

The work links the function's properties to vortex filament evolution.

Abstract

Riemann's non-differentiable function is a classic example of a continuous function which is almost nowhere differentiable, and many results concerning its analytic regularity have been shown so far. However, it can also be given a geometric interpretation, so questions on its geometric regularity arise. This point of view is developed in the context of the evolution of vortex filaments, modelled by the Vortex Filament Equation or the binormal flow, in which a generalisation of Riemann's function to the complex plane can be regarded as the trajectory of a particle. The objective of this document is to show that the trajectory represented by its image does not have a tangent anywhere. For that, we discuss several concepts of tangent vectors in view of the set's irregularity.