Variations on criteria of Pólya and Turán for the Riemann hypothesis
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)].
scientific article; zbMATH DE number 7355818
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