This is the fault of Igor Rivin, who asked about sums of divisor functions. I will put in links eventually. What I would like to know is the size of the right hand side in Ramanujan's formula (381), see original and a rendition FROM HERE in the second jpeg below. The quotes from Ramanujan are in (including some sign corrections) Nicolas and Robin, The Ramanujan Journal, Volume 1, Issue 2, June 1997, pages 119-153, Highly Composite Numbers by Srinivasa Ramanujan.
The questions are about the surprising size of the thing. For $s = 1/2$, formula 380, as a function of $N$ it is still larger than any power of $\log N$. Note that the construction "generalised superior highly composite numbers" always gives something smaller than any root $N^\alpha$ for $\alpha > 0$.
Question (A): is there a subinterval of the given $1/2 < s < 1$ for which the right-hand side of (381) has comparable growth to some $(\log N)^\beta$ for some $0 < \beta < \infty$?
Question (B): how do we get down to the really tiny growth for $s=1$, dominant term $\log \log N? $ Note that this was later improved in Robin's Criterion.
Between (330) and (332) we find $$ S_s(x) = - s \sum \; \frac{x^{\rho - s}}{\rho (\rho - s)} $$ where the sum is ``over the complex roots of $\zeta.$''
With that in mind, the right hand side of (381) is $$ | \zeta(s) | \exp \left\{ \mbox{Li} \left( (\log N)^{1-s} \right) - \frac{2s(2^{1/(2s)} -1)}{2s-1} \frac{(\log N)^{\frac{1}{2}-s}}{\log \log N} \right\} + \frac{S_s(\log N)}{\log \log N} + O \left\{ \frac{(\log N)^{\frac{1}{2}-s}}{(\log \log N)^2} \right\} $$