What kind of upper bound can one get for
$$
\sum_{d|n}\frac{1}{d^{\sigma}},
$$
where $0<\sigma<1$?  The best I can do is $\ll n^{(1-\sigma)/2}$ by breaking the sum at $\sqrt{n}$, using the symmetry of the divisors, and replacing the sum on $d|n$, $d<\sqrt{n}$, with the sum over all $d<\sqrt{n}$.  Since the hyperbola method gives that the average value is $O(1)$, I'm hoping for a tighter upper bound.