On Robin's Inequality and the Kaneko-Lagarias Inequality

Abstract

We provide new, elementary proofs that Robin's inequality and the Lagarias inequality hold for almost every number, including all numbers not divisible by one of the prime numbers $2$, $3$, $5$; all primorials; given $k$ a natural number, all sufficiently large numbers of the form $2^kn$ for $n\ge1$ odd; and all $21$-free integers. Additionally, we prove that the Kaneko-Lagarias inequality holds for all natural numbers if and only if it holds for all superabundant numbers.