Best known upper bound for $\sigma(n)$

Question

I was looking around on Wikipedia and OEIS and the best I could find relies on RH (Robin's Theorem or Lagrimas's Theorem). Are there any good known bounds that don't rely on RH?

Answer 1

From Robin we have an unconditional result (1984) that says that for $n \geq 13,$ we have the bound $$ \frac{\sigma(n)}{n} < \; e^\gamma \log \log n + \frac{0.64821364942...}{\log \log n},$$ with the constant in the numerator giving equality for $n=12.$ Note $$ e^\gamma = 1.7810724179901979852365\ldots $$

This is stated in a paper in English by Briggs as Theorem 1.1.

If you would like to experiment, the easiest sequence of numbers that still gives very large values (almost $1$) of $\frac{\sigma(n)}{n \; e^\gamma \; \log \log n}$ is $$ A_n = \operatorname{lcm} \{ 1,2,3,4,5, ..., n-1,n \}, $$ where $A_n \neq A_{n-1}$ only when $n$ is a prime or prime power. The very best values come from the sequence of Colossally Abundant numbers, but that is a more difficult computer program.

Let's see, the Riemann Hypothesis is equivalent to the statement that, for $n \geq 5041 = 1 + 7!,$ the number $0.64821364942...$ can be replaced by $0.$