Normality of algebraic numbers and the Riemann zeta function

A real number is called simply normal to base $b$ if every digit $0,1,\dots ,b-1$ should appear in its $b$-adic expansion with the same frequency $1\mathrm{/}b$. A real number is called normal to base $b$ if it is simply normal to every base $b,b^2,\dots$. In this article, we discover a relation between the normality of algebraic numbers and a mean of the Riemann zeta function on vertical arithmetic progressions. Consequently, we reveal that a positive algebraic irrational number $\alpha$ is normal to base $b$ if and only if we have $$\underset{N\to \mathrm{\infty }}{\lim }\frac{1}{\log N}\sum _{1\le \mathrm{\mid }n\mathrm{\mid }\le N}\zeta (-k+\frac{2\pi in}{\log b})\frac{e^{2\pi in\log \alpha \mathrm{/}\log b}}{n^{k+1}}=0$$ for every integer $k\ge 0$.