Preventing Exceptions to Robins InEquality
Thomas Schwabh\"auser

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 (RIE). The largest known violation of RIE is . In this paper Robin's multipliers are split into logarithmic terms and relative divisor sums . A violation of RIE above is proposed to imply oscillations that cause to exceed . 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
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Taxonomy
TopicsAnalytic Number Theory Research · Advanced Mathematical Identities · Limits and Structures in Graph Theory
