A function that satisfies $n^{f(n)}\zeta(f(n)) \to e^\gamma n\log\log n$

Question

I have been trying to find a function $f$ that yields the following:

$$n^{f(n)}\zeta(f(n)) \to e^\gamma n\log\log n$$ where $f(n)\to1$ and $f(n)\gt1$ for all sufficient $n$.

I suspect that Mertens theorems may be relevant.

It also may be useful to note that $$\lim\limits_{t\to 0}\left(1-n^{-t}\right)\zeta(1+t)=\log n.$$