On the linear independence of the set of Dirichlet exponents (Q1932494)

In the paper under review the author proves that, for every transcendental numbers \(\alpha_1,\dots,\alpha_k\) such that \(\alpha_1,\dots,\alpha_{k-1}\) are algebraically independent over \(\mathbb{Q}\) and \(\alpha_k\in \mathbb{Q}(\alpha_1,\dots,\alpha_{k-1})\) cannot be written in the form \((a\alpha_i+c)/b\) for \(i=1,\dots,k-1\), \(a,b\in\mathbb{N}\), \(c\in\mathbb{Z}\) with \(\gcd(a,b)=1\), the set of numbers \[ \begin{multlined} \{\log(n+\alpha_1):n\in\mathbb{N}_0\}\cup\{\log(n+\alpha_2):n\in\mathbb{N}_0\}\cup\dots \cup\{\log(n+\alpha_{k-1}):n\in\mathbb{N}_0\}\\ \cup\{\log(n+\alpha_k):q\leq n\in\mathbb{N}_0\} \end{multlined} \] is linearly independent over \(\mathbb{Q}\) for suitable \(q\geq 0\). The numbers \(\log(n+\alpha)\) for \(n\in\mathbb{N}_0\) are known as Dirichlet exponents and appear in the definition of the Hurwitz zeta-function \(\zeta(s;\alpha)\) as the Dirichlet series \(\sum_{n=0}^\infty \exp(-s\log(n+\alpha))\). Using this fact the author proves that the Hurwitz zeta-functions \(\zeta(s;\alpha_1),\dots,\zeta(s;\alpha_k)\) are jointly strongly universal, provided the set of numbers \[ \{\log(n+\alpha_1):q_1\leq n\in\mathbb{N}_0\}\cup\{\log(n+\alpha_2):q_2\leq n\in\mathbb{N}_0\}\cup\dots\cup\{\log(n+\alpha_{k}):q_k\leq n\in\mathbb{N}_0\} \] is linearly independent over \(\mathbb{Q}\) for some \(q_1,\dots,q_k\geq 0\).