A criterion for the simple normality of fractional powers of two via the Riemann zeta function

Abstract A real number is simply normal to base b if its base- b expansion has each digit appearing with average frequency tending to $1/b$ . In this article, we discover a relation between the frequency at which the digit $1$ appears in the binary expansion of $2^{p/q}$ and a mean value of the Riemann zeta function on vertical arithmetic progressions. In particular, we show that $$\begin{align*}\lim_{l\to \infty} \frac{1}{l}\sum_{0<|n|\leq 2^l } \zeta\left(\frac{2 n\pi i}{\log 2}\right) \frac{e^{2n\pi i p/q} }{n} =0 \end{align*}$$ if and only if $2^{p/q}$ is simply normal to base $2$ .