The Nicolas and Robin inequalities with sums of two squares

In 1984, G. Robin proved that the Riemann hypothesis is true if and only if the Robin inequality $σ(n)<e^γn\log\log n$ 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 $\cS$ of the natural numbers such that the Robin inequality holds for all but finitely many $n\in\cS$. 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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