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.

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.

Update: Thanks to Lucia for reminding me of the better but still trivial bound $$ < d(n)\ll C(\epsilon)n^\epsilon. $$ The article Will Jagy cites below shows that on the Riemann Hypothesis the sum (in the $\sigma>1/2$ case I'm interested in) is $$ \exp\left(O\left(Li(\log(n)^{1-\sigma})\right)\right). $$