Transcendency of variants of Mills' constant
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915658501980160 |
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| author | Saito, Kota |
| author_facet | Saito, Kota |
| contents | Let $\lfloor x\rfloor$ denote the integer part of $x$. For every sequence $(C_k)_{k\ge 1}$ of positive integers, we define $ξ(C_k)$ as the smallest real number $ξ>1$ such that $\lfloor ξ^{C_k} \rfloor$ is a prime number for every positive integer $k$. The number $ξ(3^k)$ is called Mills' constant. Recently, the author showed that $ξ(3^k)$ is irrational; however, the transcendency remains open. In this paper, we show that Mills' constant is transcendental under the Density Hypothesis of the Riemann zeta function. Furthermore, we obtain four classes of sequences $(C_k)_{k\ge 1}$ for which we can verify the arithmetic properties of $ξ(C_k)$. For simplicity, we give four representative examples belonging to each class: (A) $ξ(\lfloor b^k\rfloor)$ is irrational for every real number $b\ge 1+\sqrt{2}$; (B) $ξ((1+\sqrt{2})^k+(1-\sqrt{2})^k)$ is transcendental; (C) $ξ(r3^k-1)$ is transcendental for every integer $r\ge 4.003\times 10^{14}$; (D) $ξ(3^{k-\lfloor (\log k)^{1/2} \rfloor}2^{\lfloor (\log k)^{1/2}\rfloor})$ is transcendental. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_16068 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Transcendency of variants of Mills' constant Saito, Kota Number Theory 11J72, 11J81 Let $\lfloor x\rfloor$ denote the integer part of $x$. For every sequence $(C_k)_{k\ge 1}$ of positive integers, we define $ξ(C_k)$ as the smallest real number $ξ>1$ such that $\lfloor ξ^{C_k} \rfloor$ is a prime number for every positive integer $k$. The number $ξ(3^k)$ is called Mills' constant. Recently, the author showed that $ξ(3^k)$ is irrational; however, the transcendency remains open. In this paper, we show that Mills' constant is transcendental under the Density Hypothesis of the Riemann zeta function. Furthermore, we obtain four classes of sequences $(C_k)_{k\ge 1}$ for which we can verify the arithmetic properties of $ξ(C_k)$. For simplicity, we give four representative examples belonging to each class: (A) $ξ(\lfloor b^k\rfloor)$ is irrational for every real number $b\ge 1+\sqrt{2}$; (B) $ξ((1+\sqrt{2})^k+(1-\sqrt{2})^k)$ is transcendental; (C) $ξ(r3^k-1)$ is transcendental for every integer $r\ge 4.003\times 10^{14}$; (D) $ξ(3^{k-\lfloor (\log k)^{1/2} \rfloor}2^{\lfloor (\log k)^{1/2}\rfloor})$ is transcendental. |
| title | Transcendency of variants of Mills' constant |
| topic | Number Theory 11J72, 11J81 |
| url | https://arxiv.org/abs/2508.16068 |