Well assuming that the Prime Number Theorem is true, when substituting $\delta(x)$, the density of primes near $x$—which I am being vague of what it means 'cause I don't have enough foundation about the topic—is asymptotic to $\frac{1}{\log(x)}$. Then, when I graphed the difference between $\log(x!)$ & $x\cdot\sum_{n\leq x}\frac{1}{n}$ it seemed linear. What is in my mind is just the harmonic series (which is the Digamma function). Then how to go from there.
Note; how did I get the expression in the first place? $$n!=\prod_{p\leq n}p^{V(n,p)}\approx\prod_{p\leq n}p^{\frac{n}{p}}$$ Where $p$ is a prime and $V(n,p)$ is how many multiples of the prime $p$ in $n!$
My Question "Is $\displaystyle \lim_{N\to \infty}\frac{|\log(N!)-NH_N|}{N}=0$" (Where $H_n$ is $n$-th Harmonic Number )
If not then how to correct the error.