On the Least Colossally Abundant Exception to Robin's Inequality
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911236300472320 |
|---|---|
| author | Zimov, Bruce |
| author_facet | Zimov, Bruce |
| contents | Robin's Inequality posits $G(n)<e^γ$ for $n>5040$. Robin also showed that if the Riemann Hypothesis (RH) is false, then $G(n)>e^γ\left(1+\displaystyle\frac{c}{(\log n)^{b}}\right)$ for infinitely many values of $n$. By analyzing the prime or semiprime quotient $\displaystyle\frac{n}{m}$ for consecutive Colossally Abundant (CA) numbers $m$ followed by $n$ (where $m$ satisfies Robin's Inequality and $n$ violates it), we demonstrate that if the Riemann Hypothesis is false, then the least CA counterexample, $n$, must be constrained to the band $e^γ<G(n)<e^γ\left(1+\displaystyle\frac{c}{(\log n)^b}\right)$ where $0 < b < 1/2$, i.e. excluded from the infinite set beyond the higher threshold. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_23889 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Least Colossally Abundant Exception to Robin's Inequality Zimov, Bruce Number Theory 11A25, 11N37, 11N56 Robin's Inequality posits $G(n)<e^γ$ for $n>5040$. Robin also showed that if the Riemann Hypothesis (RH) is false, then $G(n)>e^γ\left(1+\displaystyle\frac{c}{(\log n)^{b}}\right)$ for infinitely many values of $n$. By analyzing the prime or semiprime quotient $\displaystyle\frac{n}{m}$ for consecutive Colossally Abundant (CA) numbers $m$ followed by $n$ (where $m$ satisfies Robin's Inequality and $n$ violates it), we demonstrate that if the Riemann Hypothesis is false, then the least CA counterexample, $n$, must be constrained to the band $e^γ<G(n)<e^γ\left(1+\displaystyle\frac{c}{(\log n)^b}\right)$ where $0 < b < 1/2$, i.e. excluded from the infinite set beyond the higher threshold. |
| title | On the Least Colossally Abundant Exception to Robin's Inequality |
| topic | Number Theory 11A25, 11N37, 11N56 |
| url | https://arxiv.org/abs/2510.23889 |