Some geometric properties of Riemann’s non-differentiable function

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 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.