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 $$\lim\limits_{t\to 0}\left(1-n^{-t}\right)\zeta(1+t)=\log n.$$

  • $\begingroup$ What is it supposed to be useful for ? Replace $\zeta(f(n))$ by $\frac{1}{f(n)-1}$, let $u(x) = x / \log(x)$ then you get something of the form $u(n^{f(n)-1}) \approx g(n)$ so that $f(n) \approx 1+\frac{\log(u^{-1}(g(n)))}{\log(n)}$. $\endgroup$ – reuns Oct 27 '18 at 21:48
  • $\begingroup$ I’m just doing experiment related to the Robin Inequality What you provided actually helped me. $f(n)/gt 1$ doesn’t hold but it helps anyway. $\endgroup$ – tyobrien Oct 28 '18 at 5:30

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