The Nicolas and Robin inequalities with sums of two squares

TL;DR

This paper investigates the Robin and Nicolas inequalities within subsets of natural numbers, especially sums of two squares, establishing conditions under which these inequalities hold and identifying exceptions.

Contribution

It characterizes subsets of natural numbers, including sums of two squares, where Robin's inequality holds, and explicitly identifies exceptions among sums of two squares.

Findings

Robin inequality holds for all but finitely many sums of two squares.

Explicitly identifies the finite set of sums of two squares violating Robin's inequality.

Establishes the equivalence of Robin and Nicolas inequalities in this context.

Abstract

In 1984, G. Robin proved that the Riemann hypothesis is true if and only if the Robin inequality $\sigma(n)<e^\gamma n\log\log n$ holds for every integer $n>5040$, where $\sigma(n)$ is the sum of divisors function, and $\gamma$ 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 $n/\phi(n)<e^{\gamma}\log \log n$; since $\sigma(n)/n<n/\phi(n)$ for $n>1$ our results for the Robin inequality follow at once.