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arXiv · 2312.13943

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

Yuya Kanado·Kota Saito
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Abstract

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 \[ \lim_{l\to \infty} \frac{1}{l}\sum_{0<|n|\leq 2^l } ζ\left(\frac{2 nπi}{\log 2}\right) \frac{e^{2nπi p/q} }{n} =0 \] if and only if $2^{p/q}$ is simply normal to base $2$.

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Yuya Kanado, Kota Saito. 2023-12-29. The simple normality of the fractional powers of two and the Riemann zeta function. https://arxiv.org/abs/2312.13943

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