Who should I ask for Robin's paper? At any rate, I want to find out if a similar result to his can be achieved with 36 instead of 12.

Question

Robin proved unconditionally that for $\ n \ge 3$ , $$ \sigma(n)<\left(e^\gamma+{\log\log12\left({\frac73}-e^\gamma \log\log12\right)\over (\log \log n)^2}\right)n \log \log n. $$ I need a similar result with$\ 36 $, but unfortunately there's no trace of the original paper online. It is reference 9 at http://en.wikipedia.org/wiki/Colossally_abundant_number

Answer 1

seems appropriate to put the first two pages as a preview. For small $n,$ Ramanujan's method gives much better bounds, the easiest are $$ \sigma(n) \leq 3 \left( \frac{n}{2} \right)^{3/2}, $$ $$ \sigma(n) \leq 12 \left( \frac{n}{6} \right)^{5/4}, $$ $$ \sigma(n) \leq 28 \left( \frac{n}{12} \right)^{7/6}. $$