Normality of algebraic numbers and the Riemann zeta function

Abstract

A real number is called simply normal to base $b$ if every digit $0,1,\ldots ,b-1$ should appear in its $b$-adic expansion with the same frequency $1/b$. A real number is called normal to base $b$ if it is simply normal to every base $b, b^2, \ldots$. 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 \[ \lim_{N\to \infty}\frac{1}{\log N} \sum_{1\leq |n|\leq N} \zeta\left(-k+\frac{2\pi i n}{\log b} \right) \frac{e^{2\pi i n \log \alpha /\log b}}{n^{k+1}} =0 \] for every integer $k\geq 0$.