Transcendency of variants of Mills' constant

TL;DR

This paper investigates the transcendental nature of Mills' constant and its variants, proving transcendence under certain hypotheses and analyzing the arithmetic properties of related sequences.

Contribution

It proves Mills' constant is transcendental assuming the Density Hypothesis of the Riemann zeta function and classifies sequences with specific arithmetic properties of their associated constants.

Findings

Mills' constant is transcendental under the Density Hypothesis.

Certain sequences yield irrational or transcendental values of (C_k).

Four classes of sequences with verified arithmetic properties are identified.

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.