  With the aim of quantifying turbulent behaviors of vortex filaments, we study
the multifractality and intermittency of the family of generalized Riemann's
non-differentiable functions \begin{equation} R_{x_0}(t) = \sum_{n \neq 0}
\frac{e^{2\pi i ( n^2 t + n x_0 ) } }{n^2}, \qquad x_0 \in [0,1].
\end{equation} These functions represent, in a certain limit, the trajectory of
regular polygonal vortex filaments that evolve according to the binormal flow.
When $x_0$ is rational, we show that $R_{x_0}$ is multifractal and intermittent
by completely determining the spectrum of singularities of $R_{x_0}$ and
computing the $L^p$ norms of its Fourier high-pass filters, which are analogues
of structure functions. We prove that $R_{x_0}$ has a multifractal behavior
also when $x_0$ is irrational. The proofs rely on a careful design of
Diophantine sets that depend on $x_0$, which we study by crucially using the
Duffin-Schaeffer theorem and the Mass Transference Principle.
