# Riemann's non-differentiable function: a turbulent history

# Riemann's non-differentiable function: a turbulent history

Around 1860, Riemann challenged the beliefs of the time that a continuous function must have a derivative. The function he devised was not given by the typical closed expression, but rather by the infinite sum

$$R(x) = \sum_{n=1}^\infty \frac{\sin{ ( 2 \pi n^2 x ) }}{n^2}, $$

which he claimed to be continuous everywhere but to have a derivative nowhere. Claim is the correct word, for he did not prove it, leaving with it two long-lasting footprints: a big problem, unsolved until 1970, and a revolution and the consequent establishments of the foundations of modern mathematics. This function came to be known as Riemann's non-differentiable function which, turns out, is \textit{almost} nowhere differentiable. Recently, this rich analytic structure has proved itself valuable in the setting of turbulence, one of the biggest open problems in mathematical physics. I will begin the talk with a broad historic overview to then describe the structure of the function and the results that were progressively proved for it. After that, I will aim at the role that it plays in turbulence, introducing elements like vortex filaments, multifractality and intermittency, together with the relevant mathematical tools used in their study.