Small perturbations of the Riemann zeta function and their zeros
type: Item
instance of: scholarly article
title: Small perturbations of the Riemann zeta function and their zeros
zbMATH Open document ID: 1051.11046
DOI: 10.1007/BF03321061
author: E. S. Zeron
author: P. M. Gauthier
published in: Computational Methods and Function Theory
publication date: 2004-09-28T00:00:00Z
review text: The Riemann zeta-function \(\zeta(s)\) has the following well-known properties: (i) Its only pole is a simple pole at \(s=1\) with residue \(1\); (ii) \(\zeta(s)= \overline{\zeta(\overline{s})}\) for \(s\neq 1\); (iii) The functional equation. \textit{L. D. Pustyl'nikov} [Usp. Mat. Nauk 58, No. 1, 175--176 (2003)] showed that there exist functions which satisfy these three properties (with the same functional equation as that of \(\zeta(s)\)) and which uniformly approximate \(\zeta(s)\) on compact sets excluding the point \(1\), but for which the Riemann Hypothesis fails. This paper provides significant improvements on Pustyl'nikov's result. The aspects of the improvements are about where to place the new non-trivial zeros and moreover infinitely many of them, and about the sense of the approximation and the size of the sets where the improved approximations hold. A closed subset \(X\) of the complex plane is defined to be a set of tangential approximation if for every continuous function \(f : X \to \mathbb{C}\) which is holomorphic in the interior of \(X\) and for each strictly positive continuous function \(\varepsilon : X \to \mathbb{R}\), there exists an entire function \(h\) such that \(| f(z)-h(z)| < \varepsilon(z)\) for every \(z\in X\). The closed sets of tangential approximation have been completely characterized, and this paper contains examples of such sets, one being the union of the real line, the critical line and a closed disk of positive radius around \(s=1/2\). Let \(X\) be a closed subset of tangential approximation, and \(A\) be a discrete (possibly infinite) subset of \(\mathbb{C}\) such that \(A\cap (X \cup \{0,1\}) = \emptyset\). Suppose also that \(X\) and \(A\) are doubly symmetric (a set \(S\) is called doubly symmetric when \(\overline{z},\, 1-z,\, 1-\overline{z}\) are in \(S\) for each \(z\in S\)). The main theorem of this paper asserts that for each strictly positive continuous function \(\varepsilon : X \to \mathbb{R}\), there exists a …
Mathematics Subject Classification ID: 11M26
Mathematics Subject Classification ID: 30C62
Mathematics Subject Classification ID: 30E10
zbMATH DE Number: 2104120
zbMATH Keywords: Riemann zeta-function
zbMATH Keywords: approximations
zbMATH Keywords: Riemann hypothesis
reviewed by: C. Y. Yildirim
MaRDI profile type: Publication
cites work: Q3752617
cites work: Complex Approximation and Simultaneous Interpolation on Closed Sets
cites work: Approximation by Meromorphic Functions with Mittag-Leffler Type Constraints
cites work: Q3735790
full work available at URL: https://doi.org/10.1007/bf03321061
OpenAlex ID: W2088064674
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description: scientific article; zbMATH DE number 2104120