The Nicolas and Robin inequalities with sums of two squares

Details The Nicolas and Robin inequalities with sums of two squares The Nicolas and Robin inequalities with sums of two squares

2007-10-12

arxiv.org/abs/0710.2424v1

Mathematics

Number Theory

21 pages

Scientific paper

In 1984, G. Robin proved that the Riemann hypothesis is true if and only if the Robin inequality $\sigma(n)5040$, 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 $n/\phi(n)1$ our results for the Robin inequality follow at once.

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