Riemann's non-differentiable function: a physical and geometric description

It is believed that in the 1860s Riemann proposed the function \[ R(x) = \sum_{n=1}^{\infty}\frac{\sin(n^2 x)}{n^2}, \qquad x \in \mathbb{R}, \] as an example of a continuous but nowhere differentiable function, but no proof was given at the time. Since then, many authors have worked on its analytic properties, first trying to solve Riemann's guess (proven a century later, in 1970) and studying further regularity properties afterwards.

However, recent results suggest that the generalisation of Riemann's function \[ \phi(x) = \sum_{k \in \mathbb{Z}}\frac{e^{-4\pi^2ik^2x}-1}{-4\pi^2k^2}, \qquad x \in \mathbb{R} \] is a surprisingly precise approximation to a temporal trajectory in an experiment concerning vortex filaments. It can be argued then that it has also an intrinsic physical and geometric structure.

In this talk, I will present my work concerning geometric properties of $\phi(\mathbb{R})$. We will see that its Hausdorff dimension is upper bounded by 4/3. %, while to prove lower bound strictly larger than 1 is an interesting but tougher challenge. Also, motivated by the fact that tangent vectors should be the representatives of the velocity of a particle following the physical trajectory, we will see that no such tangents exist in this case. Finally, adapting Frisch's definition of intermittency and flatness, we will see that Riemann's function is intermittent, albeit in a weak form.