Robin's Inequality at Colossally Abundant Numbers and Nicolas' Inequality at Their Largest Prime Factor

Abstract

Robin's criterion states that the Riemann Hypothesis holds if and only if $\sigma(n) < e^{\gamma} \cdot n \cdot \log\log n$ for every $n > 5040$, and Nicolas' criterion states that it holds if and only if $c(q) = \log(q\#/\varphi(q\#)) - \gamma - \log\log\theta(q) > 0$ for every prime $q$, where $q\#$ is the primorial of $q$ and $\theta$ is the Chebyshev function. Let $N$ be a colossally abundant number, let $\xi(N)$ be the real number associated with $N$ by Nicolas, and let $q$ be the largest prime not exceeding $\xi(N)$. Writing $\Delta(N) = \log(\sigma(N)/N) - \gamma - \log\log\log N$, we prove without any hypothesis that, whenever $\log q \geq 31$, $1.353/(\sqrt{q} \log^{2} q) \leq \Delta(N) - c(q) + 2\sqrt{2}/(\sqrt{q} \log q) \leq 3.970/(\sqrt{q} \log^{2} q)$. Thus Robin's inequality at $N$ and Nicolas' inequality at $q$ differ by $2\sqrt{2}/(\sqrt{q} \log q)$ up to an explicit error of smaller order. As consequences, every colossally abundant counterexample to Robin's inequality yields a prime $q$ with $c(q) > 2.70/(\sqrt{q} \log q)$; conversely, every prime $q$ with $\log q \geq 31$ and $c(q) > 2\sqrt{2}(1 - 0.4787/\log q)/(\sqrt{q} \log q)$ yields a colossally abundant counterexample, so that Nicolas' oscillation theorem implies Robin's theorem on colossally abundant counterexamples; and under the Riemann Hypothesis every colossally abundant number with $\log q \geq 31$ satisfies $\Delta(N) < -0.654/(\sqrt{q} \log q)$, in accordance with the constant $2(\sqrt{2}-1)$ of Ramanujan's bound. For superabundant counterexamples, using the exponent bound of Nazardonyavi and Yakubovich, which gives $\log n \geq \psi(p_{k})$ for the largest prime factor $p_{k}$, we prove $c(p_{k}) > 0.9991/(\sqrt{p_{k}} \log p_{k})$, and we show that the least colossally abundant number not smaller than the least counterexample $n$ is itself a counterexample $N$ with $\log N - \log\xi(N) \leq \log n$.

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