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.
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