The simple normality of the fractional powers of two and the Riemann zeta function
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866913180972744704 |
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| author | Kanado, Yuya Saito, Kota |
| author_facet | Kanado, Yuya Saito, Kota |
| contents | 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 $2$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2312_13943 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The simple normality of the fractional powers of two and the Riemann zeta function Kanado, Yuya Saito, Kota Number Theory 11K16, 11M06 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 $2$. |
| title | The simple normality of the fractional powers of two and the Riemann zeta function |
| topic | Number Theory 11K16, 11M06 |
| url | https://arxiv.org/abs/2312.13943 |