Is there an upper bound (better than Robin's inequality) to the sum of divisors of non-highly-abundant numbers? I'm looking for $ f(n) $ such that $ \sigma(n) \le f(n) < ne^\gamma \log \log n $, with $ n $ not highly abundant. I'd like a proof as well. 
I hope the question is well formatted, I'm posting with my mobile. Also, I'm reposting the question because it was previously deleted, and I honestly don't know why, given that I just couldn't find it when I checked. Apologies if there's any problem.

Unlikely that there is anything useful. The trouble is that we can take any highly abundant number, even colossally abundant, which is therefore of the form $$ 2^{e_2} 3^{e_3}  5^{e_5} \cdots p_r^2 p_{r+1}^2 \cdots p_{R}^2 \cdot p_{R+1} \cdot p_{R+2} \cdots p_N    $$ 
with $e_2 \geq e_3 \geq e_5...,$
and simply replace $p_N$ by the next prime, $p_{N+1}.$ The result is no longer highly abundant, as the basic requirement for highly abundant numbers is that there be no primes skipped in the factorization, and that the exponents are non-increasing. 
I should point out that, for people experimenting with the Riemann hypothesis, there is some skill needed to correctly find large colossally abundant numbers (in proper order) by computer; it took the guy at the MO question three days to get it right. In comparison, the criterion of Nicolas uses just primorial numbers, which are far easier to work with. See Euler's Phi Function Worst Case and references there 
