  A real number is called 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 that the digit $1$
appears in the binary expansion of $2^{p/q}$ and a mean value of the Riemann
zeta function on arithmetic progressions. As a consequence, we show that \[
\lim_{l\to \infty} \frac{1}{l}\sum_{0<|n|\leq 2^l } \zeta\left(\frac{2 n\pi
i}{\log 2}\right) \frac{e^{2n\pi i p/q} }{n} =0 \] if and only if $2^{p/q}$ is
simply normal to base $2$.
