Given that $n$ is an highly abundant number, what is a function $f(x)$ that is a tight upper bound on the ratio: $$\frac{\sigma_1(n)}{n}$$ where $\sigma_1(n)$ is the sum of divisors of $n$ (including $n$ itself). I also want to know an upper bound if $n$ is a superabundant number
$f(x)$ doesn't have to reach equality on certain numbers, but it would be optimal if it did.
Best Answer
umm. For $n \geq 16,$ we get $\log \log n > 1, $ using log base $e = 2.71828...$ Then for $n \geq 16,$ $$ \frac{\sigma(n)}{n \log \log n} $$ does have an upper bound.
From Robin we have an unconditional result (1984) that says that your ratio of interest is, for $n \geq 13,$ smaller than $$ \frac{\sigma(n)}{n \log \log n} < e^\gamma + \frac{0.64821364942...}{(\log \log n)^2},$$ with the constant in the numerator giving equality for $n=12.$ Note $\gamma \approx 0.5772156649,$ while $e^\gamma \approx 1.7810724$
In particular, for $n \geq 268,$ we get $ \frac{\sigma(n)}{n \log \log n} < 2$ Below are the ratios bigger than $1.9$ with $n \geq 3.$