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We make a connection between a famous analytical object introduced in the1860s by Riemann, as well as some variants of it, and a nonlinear geometricPDE, the binormal curvature flow. As a consequence this analytical object has anon-obvious nonlinear geometric interpretation. We recall that the binormalflow is a standard model for the evolution of vortex filaments. We prove theexistence of solutions of the binormal flow with smooth trajectories that areas close as desired to curves with a multifractal behavior. Finally, we showthat this behavior falls within the multifractal formalism of Frisch andParisi, which is conjectured to govern turbulent fluids.

Riemann's non-differentiable function and the binormal curvature flow