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A real number is called simply normal to base $b$ if its base-$b$ expansionhas each digit appearing with average frequency tending to $1/b$. In thisarticle, we discover a relation between the frequency that the digit $1$appears in the binary expansion of $2^{p/q}$ and a mean value of the Riemannzeta function on arithmetic progressions. As a consequence, we show that \[\lim_{l\to \infty} \frac{1}{l}\sum_{0<|n|\leq 2^l } \zeta\left(\frac{2 n\pii}{\log 2}\right) \frac{e^{2n\pi i p/q} }{n} =0 \] if and only if $2^{p/q}$ issimply normal to base $2$.

The simple normality of the fractional powers of two and the Riemann zeta function