Strange limit: $\lim_{n\to \infty} \left(n-{\ln(n!)\over \sqrt[n]{n^k\ln2 \ln3\ln4\cdots\ln n}}\right)=2k$

Question

Observing this limit $$\lim_{n\to \infty} \left(n-{\ln(n!)\over \sqrt[n]{n^k\ln2 \ln3\ln4\cdots\ln n}}\right)=2k$$

I did try a few quite large values of n; the limit seem to converge, but slow.

Can this limit be correct and how can I shows it?

Answer 1

Let me provide some computation that concludes the question. With a bit of effort, one can show that there exists a constant $C > 0$ satisfying

$$ n - \frac{\log(n!)}{(n^k \prod_{j=2}^{n} \log j)^{1/n}} \leq - \operatorname{li}(n) - n \log\left(1 - \frac{1}{\log n}\right) + k \log n + \log\log n + C $$

for all $n \geq 2$ and $k \in \mathbb{R}$, where $\operatorname{li}(x)$ is the logarithmic integral function. Then it is not hard to check that this bound behaves like $-n^{1-o(1)}$ and hence the limit diverges.