On the increase of Gr\"onwall function value at the multiplication of its argument by a prime

TL;DR

This paper investigates the behavior of the Gr"onwall function when multiplied by a prime, establishing conditions for its increase and exploring implications related to Robin's inequality through numerical experiments with superabundant numbers.

Contribution

It introduces a condition on the Gr"onwall function that guarantees an increase upon multiplication by a prime, linking it to Robin's inequality and providing numerical insights.

Findings

Condition on n ensures G(np)>G(n) for some prime p

Numerical experiments with superabundant numbers support theoretical results

Insights into the behavior of the Gr"onwall function related to prime multiplication

Abstract

We consider the function $G(n)=\frac{\sigma(n)}{n\log\log n}$ (where $\sigma(n)=\sum_{d|n}d$) and set an imposed condition on its argument $n$, the fulfillment of which is sufficient for the existence of a prime $p$, at which $G(np)>G(n)$. This inequality is of interest in connection with the Robin's inequality. The paper also presents the results of numerical experiment conducted with superabundant numbers.