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

We investigate the variational properties of Riemann’s ‘‘nondifferentiable’’ function $R$. We show that $R$ has finite $p$-variation for $p>4\mathrm{/}3$ and infinite $p$-variation for $p<4\mathrm{/}3$. We also conjecture that the critical $4\mathrm{/}3$-variation of $R$ is finite, and make partial progress towards a proof of this conjecture. While the cases $p\mathrm{\ne }4\mathrm{/}3$ are handled using ‘‘soft’’ methods, the critical case $p=4\mathrm{/}3$ requires a more careful analysis, drawing on work of Duistermaat together with a novel application of the Farey sequence. 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.