Abstract
In 1984, G. Robin proved that the Riemann hypothesis is true if and only if the Robin inequality σ(n) < e γ n log log n holds for every integer n > 5040, where σ(n) is the sum of divisors function, and γ is the Euler–Mascheroni constant. We exhibit a broad class of subsets \({\mathcal {S}}\) of the natural numbers such that the Robin inequality holds for all but finitely many \({n \in \mathcal {S}}\) . As a special case, we determine the finitely many numbers of the form n = a 2 + b 2 that do not satisfy the Robin inequality. In fact, we prove our assertions with the Nicolas inequality n/φ(n) < e γ log log n; since σ(n)/n < n/φ(n) for n > 1 our results for the Robin inequality follow at once.
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Communicated by U. Zannier.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Banks, W.D., Hart, D.N., Moree, P. et al. The Nicolas and Robin inequalities with sums of two squares. Monatsh Math 157, 303–322 (2009). https://doi.org/10.1007/s00605-008-0022-x
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DOI: https://doi.org/10.1007/s00605-008-0022-x