Normality of algebraic numbers and the Riemann zeta function
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912159072518144 |
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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 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 $α$ is normal to base $b$ if and only if we have \[ \lim_{N\to \infty}\frac{1}{\log N} \sum_{1\leq |n|\leq N} ζ\left(-k+\frac{2πi n}{\log b} \right) \frac{e^{2πi n \log α/\log b}}{n^{k+1}} =0 \] for every integer $k\geq 0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_02337 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Normality of algebraic numbers and the Riemann zeta function Kanado, Yuya Saito, Kota Number Theory 11K16, 11M06 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 $α$ is normal to base $b$ if and only if we have \[ \lim_{N\to \infty}\frac{1}{\log N} \sum_{1\leq |n|\leq N} ζ\left(-k+\frac{2πi n}{\log b} \right) \frac{e^{2πi n \log α/\log b}}{n^{k+1}} =0 \] for every integer $k\geq 0$. |
| title | Normality of algebraic numbers and the Riemann zeta function |
| topic | Number Theory 11K16, 11M06 |
| url | https://arxiv.org/abs/2412.02337 |