  We make a connection between a famous analytical object introduced in the
1860s by Riemann, as well as some variants of it, and a nonlinear geometric
PDE, the binormal curvature flow. As a consequence this analytical object has a
non-obvious nonlinear geometric interpretation. We recall that the binormal
flow is a standard model for the evolution of vortex filaments. We prove the
existence of solutions of the binormal flow with smooth trajectories that are
as close as desired to curves with a multifractal behavior. Finally, we show
that this behavior falls within the multifractal formalism of Frisch and
Parisi, which is conjectured to govern turbulent fluids.
