Riemann's non-differentiable function is intermittent

Riemann's non-differentiable function, introduced in the middle of the 19th century as a purely mathematical pathological object, is relevant in the study of the binormal flow, as shown recently by De La Hoz and Vega. From this physical point of view, the function is therefore related to turbulent phenomena. We rigorously study the fine intermittent nature of this function on small scales. To do so, we define the flatness, an analytic quantity measuring it, in two different ways: one in the physical space and the other one in the Fourier space. We prove that both expressions diverge logarithmically as the relevant scale parameter tends to 0. The regularity of Riemann's non-differentiable function is a classical subject, heavily linked to its small-scale behaviour. However, our subtle asymptotics for a classical hydrodynamical quantity are new and sharp