# Some geometric properties of Riemann's non-differentiable function

## TL;DR

This paper investigates the geometric properties of Riemann's non-differentiable function's complex image, establishing its Hausdorff dimension bound and the absence of tangents, linking it to vortex filament experiments.

## Contribution

It proves that the Hausdorff dimension of the function's complex image is at most 4/3 and shows it has no tangent at any point, connecting geometric analysis with physical models.

## Key findings

- Hausdorff dimension of the image ≤ 4/3
- The image has no tangent at any point
- Supports connection to vortex filament experiments

## Abstract

Riemann's non-differentiable function is a celebrated example of a continuous but almost nowhere differentiable function. There is strong numeric evidence that one of its complex versions represents a geometric trajectory in experiments related to the binormal flow or the vortex filament equation. In this setting, we analyse certain geometric properties of its image in $\mathbb{C}$. The objective of this note is to assert that the Hausdorff dimension of its image is no larger than 4/3 and that it has nowhere a tangent.