On the linear independence of the set of Dirichlet exponents

Abstract

Given k ≥ 2 let α₁, ..., α_k be transcendental numbers such that α₁, ..., α_k–1 are algebraically independent over Q and α_k $\in$ Q(α₁, ..., α_k–1), but α_k, ≠ (aα_i + c)/b for some i $\in$ {1, ..., k – 1} and some a, b $\in$ N, c $\in$ Z satisfying gcd(a,b) = 1. We prove that then there exists a nonnegative integer q such that the set of so-called Dirichlet exponents log(n + α_j, where n runs through the set of all nonnegative integers for j = 1, ..., k – 1 and n = q, q + 1, q + 2, ... for j = k, is linearly independent over Q. As an application we obtain a joint universality theorem for corresponding Hurwitz zeta functions ζ(s, α₁), ..., ζ(s, α_k) in the strip {s $\in$ C: 1/2 < $\Re$(s) < 1}. In our approach we follow a recent result of Mishou who analyzed the case k = 2.