Small perturbations of the Riemann zeta function and their zeros
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 meromorphic function \(\zeta_{\nu}(s)\) satisfying the three properties given above, and with \(| \zeta(s)-\zeta_{\nu}(s)| <\varepsilon(s)\) for every \(s\in X\). Moreover, \(\zeta_{\nu}(s)\) vanishes on \(A\), and \(\zeta(s)\) and \(\zeta_{\nu}(s)\) have the same zeros in the complement of \(A\). The other theorem of this paper shows the existence of quasi-meromorphic functions agreeing with \(\zeta(s)\) outside of an even smaller set, and approximates \(\zeta(s)\) everywhere, but for which the Riemann Hypothesis fails.
scientific article; zbMATH DE number 2104120
scientific article