Variations on criteria of Pólya and Turán for the Riemann hypothesis
type: Item
instance of: scholarly article
title: Variations on criteria of Pólya and Turán for the Riemann hypothesis
zbMATH Open document ID: 1500.11064
author: Emre Alkan
published in: Journal of Number Theory
publication date: 2021-06-07T00:00:00Z
review text: Let \(\lambda(n)\) be the Liouville function, defined by \[ \lambda(n):=(-1)^{\Omega(n)}, \] where \(\Omega(n)\) is the total number of prime divisors of \(n\). It is well-known that the Riemann hypothesis for the Riemann zeta function is equivalent to the validity of the bound \[ \sum\limits_{n\le x} \lambda(n)\ll x^{1/2+\varepsilon} \quad \mbox{as } x\rightarrow \infty \] for all \(\varepsilon>0\). There are other criteria for the Riemann hypothesis in terms of the Liouville function. For \(0\le \alpha\le 1\), let \[ L_{\alpha}(x) := \sum\limits_{n\le x} \frac{\lambda(n)}{n^{\alpha}}. \] Pólya and Turán demonstrated that the eventual sign constancy of each of \(L_0(x)\) and \(L_1(x)\) alone implies the Riemann hypothesis. This sign constancy was, however, disproved Haselgrove. The article under review provides a remedy for this. It is shown that the eventual negativity of each of \(L_{\alpha}(x)\), \(L_{\alpha}'(x)\), \(L_{\alpha}''(x)\) for all \(\alpha \in (1/2,1)\) is equivalent to the Riemann hypothesis. This is extended to a criterion for the quasi Riemann hypothesis RH\((\rho)\) which claims that there is no zero of the Riemann zeta function with real part greater than \(\rho\). More precisely, it is shown that RH\((\rho)\) holds if and only if \(L_{\alpha}(x)\) or \(L_{\alpha}'(x)\) or \(L_{\alpha}''(x)\) is eventually negative for all \(\alpha\in (\rho,1)\). Similar criteria are developed for corresponding sums involving the Möbius function. These results are further extended to sums over semigroups of positive integers obtained by sifting out all integers divisible by a subset of prime numbers subject to specific growth conditions. The methods in this article are inspired by work of \textit{M. J. Mossinghoff} and \textit{T. S. Trudgian} [J. Aust. Math. Soc. 93, No. 1--2, 157--171 (2012; Zbl 1305.11076)].
reviewed by: Stephan Baier
Mathematics Subject Classification ID: 11M26
Mathematics Subject Classification ID: 30B50
zbMATH DE Number: 7355818
zbMATH Keywords: Liouville function
zbMATH Keywords: Möbius function
zbMATH Keywords: Riemann hypothesis
zbMATH Keywords: quasi-Riemann hypothesis
zbMATH Keywords: Pólya's criterion
zbMATH Keywords: Turán's criterion
Wikidata QID: Q114157101
MaRDI profile type: Publication
full work available at URL: j.jnt.2021.01.004
OpenAlex ID: W3132358529
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DOI: 10.1016/J.JNT.2021.01.004
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description: scientific article; zbMATH DE number 7355818