On the $p$-variation of Riemann's "nondifferentiable'' function

Abstract

We investigate the variational properties of Riemann&#39;s &#39;&#39;nondifferentiable&#39;&#39; function $R$. We show that $R$ has finite $p$-variation for $p>4/3$ and infinite $p$-variation for $p<4/3$. As an application of our results, we provide a new look on an upper estimate of the Hausdorff dimension of the image of $R$, due to Eceizabarrena.