Useful functions between polylogarithmic and polynomial?

Question

I'm wondering if there are any useful functions asymptotically greater than a polylogarithmic function and less than a polynomial function.

That is, a function $f(n)$ such that

$f(n) = \omega(\log(n)^k)$ for some constant $k > 0$

and

$f(n) = o(n^k)$ for some constant $k > 0$

What I mean by useful, is that it was used in a proof, algorithm, etc. rather than simply producing a function to fit these restrictions.

Answer 1

According to Wikipedia (which attributes the following result to Knuth), the running time of the mixed-level Toom–Cook algorithm for integer multiplication is $$ \Theta(n\log n \cdot 2^{\sqrt{2\log n}}). $$ The function $2^{\sqrt{2\log n}}$ is super-polylogarithmic but sub-polynomial.