A joint universality theorem for periodic Hurwitz zeta-functions. II.
The aim of this paper is to prove a joint universality theorem for the functions \(\zeta(s, \alpha_1; \mathfrak A_1), \ldots,\) \(\zeta(s, \alpha_r; \mathfrak A_r)\) without using the hypothesis on the rank of the matrix \(A\) [\textit{A. Javtokas} and the first author [Bull. Aust. Math. Soc. 78, No. 1, 13--33 (2008; Zbl 1228.11136)]. Let \[ L(\alpha_1, \ldots, \alpha_r) =\{\log(m + \alpha_j ): j = 1, \ldots, r, m\in\mathbb N_0.\} \] Theorem. Suppose that the set \(L(\alpha_1, \ldots, \alpha_r)\) is linearly independent over \(\mathbb Q\). Then the functions \(\zeta(s, \alpha_1; \mathfrak A_1), \ldots, \zeta(s, \alpha_r; \mathfrak A_r)\) are universal. The authors note that the linear independence of \(L(\alpha_1, \ldots, \alpha_r)\) holds whenever the numbers \(\alpha_1, \ldots, \alpha_r\) are algebraically independent over \(\mathbb Q\). Therefore, the hypothesis on the numbers \(\alpha_1, \ldots, \alpha_r\) in the present Theorem is weaker than that of the preceding paper (loc. cit.).
scientific article; zbMATH DE number 5626714