Approximating all meromorphic functions by linear motions of the Riemann zeta-function
type: Item
instance of: scholarly article
title: Approximating all meromorphic functions by linear motions of the Riemann zeta-function
zbMATH Open document ID: 1264.30045
DOI: 10.1007/BF03321841
published in: Computational Methods and Function Theory
publication date: 2013-01-11T00:00:00Z
review text: 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.
reviewed by: Augustin Mouze
Mathematics Subject Classification ID: 30K10
Mathematics Subject Classification ID: 30E10
Mathematics Subject Classification ID: 41A30
Mathematics Subject Classification ID: 11M99
zbMATH DE Number: 6124800
zbMATH Keywords: Voronin universality
zbMATH Keywords: approximation in the complex domain
zbMATH Keywords: approximation by special functions
author: P. M. Gauthier
MaRDI profile type: Publication
cites work: Lavrentiev's approximation theorem with nonvanishing polynomials and universality of zeta-functions
cites work: Non-normal sequences of holomorphic functions and universality
cites work: Approximation of and by the Riemann zeta-function
cites work: Approximating functions by the Riemann zeta-function and by polynomials with zero constraints
cites work: Value distribution of \(L\)-functions
description: scientific article; zbMATH DE number 6124800