bibkey: elima2022robin
authors: Emmanuel Elima
year: 2022
title: On Robin's inequality for positive integers and related bounds
doi: null
url: https://easychair.org/publications/preprint/xW8q
claim: Robin's inequality holds for every integer above 5040 with at most twelve distinct prime divisors, by reduction to 21-full integers.
strata_touched: []
license: citation-only
triage: anchor

On Robin’s inequality for positive integers and related bounds

It is a preprint; this card records the stated interfaces and does not audit the proofs.

Write $\omega (n)=k$ for the number of distinct prime divisors of $n$.

• Lemma 1.4 states that $\sigma (n)<e^{\gamma }n\log \log n$ holds whenever $k\le 12$. Its proof assumes $n$ is 21-full, using the 21-free case of Axler’s theorem, so that some prime $q$ has $q^{21}\mid n$ and $n\ge 2^{21}{\prod }_{i=2}^k{p}_i$. It then checks ${\prod }_{i\le k}{p}_i/(p_i-1)<e^{\gamma }\log \log \left(2^{21}{\prod }_{i=2}^k{p}_i\right)$ for $k\le 12$ by calculation.
• Theorem 1.2 states the inequality for every $t$-full $n$ with $t>0.44k\log (k\log k)+0.49k+1$.
• Theorem 1.3 states it for every $n$ with $\log p_k(1+1/{\log }^2{p}_k)<\log \log n$, where $p_k$ is the $k$-th prime.

The bounded-$\omega$ statement therefore rests on the cited 21-free theorem and on the author’s calculation; neither is a Lean verification.

Verified locator

• URL: https://easychair.org/publications/preprint/xW8q
• Version and location: EasyChair Preprint No. 9446, dated 11 December 2022; Lemma 1.4, Theorem 1.2 and Theorem 1.3.