Transcendency of variants of Mills' constant

Abstract

Let $\lfloor x\rfloor$ denote the integer part of $x$. For every sequence $(C_k)_{k\ge 1}$ of positive integers, we define $\xi(C_k)$ as the smallest real number $\xi>1$ such that $\lfloor \xi^{C_k} \rfloor$ is a prime number for every positive integer $k$. The number $\xi(3^k)$ is called Mills' constant. Recently, the author showed that $\xi(3^k)$ 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 $\xi(C_k)$. For simplicity, we give four representative examples belonging to each class: (A) $\xi(\lfloor b^k\rfloor)$ is irrational for every real number $b\ge 1+\sqrt{2}$; (B) $\xi((1+\sqrt{2})^k+(1-\sqrt{2})^k)$ is transcendental; (C) $\xi(r3^k-1)$ is transcendental for every integer $r\ge 4.003\times 10^{14}$; (D) $\xi(3^{k-\lfloor (\log k)^{1/2} \rfloor}2^{\lfloor (\log k)^{1/2}\rfloor})$ is transcendental.