Approximating all meromorphic functions by linear motions of the Riemann zeta-function
In 1975 Voronin showed that the translates of the Riemann zeta function \(\zeta\) possess a remarkable universal property concerning the approximation of a given analytic function. This spectacular result was extended by \textit{B. Bagchi} [Math. Z. 181, 319--334 (1982; Zbl 0479.10028)]. In the paper under review, using Bagchi's theorem, the author proves that the zeta-function \(\zeta\) satisfies the following property: ``for each function \(g\) meromorphic on \(\mathbb{C}\), there exists a sequence \(q_n(z)=a_n+b_nz\) of linear transformations and an increasing sequence of compact sets \(K_m\) whose union is the complex plane \(\mathbb{C}\) such that the sequence \((\zeta\circ q_n)\) converges spherically uniformly to \(g\) on each \(K_m\)'' (quotation from the abstract). Observe that the exhaustion \(K_m\) depends on the function \(g\). Finally, the author obtains a similar result by replacing the Riemann zeta function by Dirichlet series which belong to the Steuding class.
scientific article; zbMATH DE number 6124800