Preventing Exceptions to Robins InEquality

TL;DR

This paper investigates Robin's inequality for large numbers, proposing that violations above a certain point imply oscillations causing specific divisor sum terms to surpass logarithmic terms, using conjectures related to CA numbers.

Contribution

It introduces a novel split of Robin's multipliers into logarithmic and divisor sum components and explores implications of potential violations using conjectures on CA numbers.

Findings

Violations above n_8 suggest oscillations in divisor sums.

Robin's multipliers can be decomposed into logarithmic and divisor sum parts.

The paper proposes conditions under which exceptions to Robin's inequality can be prevented.

Abstract

For sufficiently large n Ramanujan gave a sufficient condition for the truth Robin's InEquality $X(n):=\frac{\sigma(n)}{n\ln\ln n}<e^{\gamma}$ (RIE). The largest known violation of RIE is $n_8=5040$. In this paper Robin's multipliers are split into logarithmic terms $\mathcal{L}$ and relative divisor sums $\mathcal{G}$. A violation of RIE above $n_{8}$ is proposed to imply oscillations that cause $\mathcal{G}$ to exceed $\mathcal{L}$. To this aim Alaoglu and Erd\H{o}s's conjecture for the CA numbers algorithm is used and the paper's key points are in section 4.2