How to evaluate double limit of multifactorial $\lim\limits_{k\to\infty}\lim\limits_{n\to 0} \sqrt[n]{n\underbrace{!!!!\cdots!}_{k\,\text{times}}}$

Solution 1:

Proposition:

$$\boxed{\;\ln m=-\gamma\ln2+\frac{\ln^22}{2}\;}$$

Explanation (sketch of the proof): for $i,k\to\infty$ such that $i/k=x$ with finite $x>0$ the expression under the sum $\sum_{i=1}^{k-1}$ tends to $0$ and therefore this range of $i$ does not contribute to the limit. We may therefore restrict the summation to the values with $i/k\ll 1$. Expand everything in $i/k$: $$\cot\frac{\pi i }{k}\to \frac{k}{\pi i },\qquad \ln\frac{\Gamma(i/k)}{k}\to -\ln i,\qquad \ln k^{i/k}=i\frac{\ln k}{k}.$$ The third piece should not contribute to the limit $k\to\infty$ because of the cutting factor $\frac{\ln k}{k}$. The first two produce the following expression in the exponential: $$\sum_{i=1}^\infty (-1)^{i+1}\frac{\ln i}{i}=-\gamma\ln2+\frac{\ln^22}{2}.$$ There admittedly remain some details to be filled for the above to become a rigorous proof, however I leave it to whoever is interested.