Transcendency of variants of Mills' constant

Let x denote the integer part of x. For every sequence (Cₖ)k≥ 1 of positive integers, we define ξ(Cₖ) as the smallest real number $ξ>1$ such that ξCₖ is a prime number for every positive integer k. The number ξ(3ᵏ) is called Mills' constant. Recently, the author showed that ξ(3ᵏ) is irrational, but the transcendency of this number is still open. In this paper, we give sufficient conditions on (Cₖ)k≥ 1 to determine the transcendency of ξ(Cₖ). For example, we obtain the following three results: (A) ξ( bᵏ) is irrational for every real number b≥ 1+√2; (B) ξ((1+√2)ᵏ+(1-√2)ᵏ) is transcendental; (C) ξ(r3ᵏ-1) is transcendental for every integer r≥ 4.003× 10¹⁴. Furthermore, assuming the Riemann hypothesis, ξ(3ᵏ-1) is transcendental. However, we still do not know whether Mills' constant is transcendental.