Stirling's formula and primes

Stirling. $\ln(n!)\sim n\ln n-n$.

Prime factorization. $n!=\prod_{p\le n}p^{v_p(n!)}$ with $v_p(n!)=\sum_k\lfloor n/p^k\rfloor$.

Estimate. $\ln(n!)=\sum_{p\le n}v_p(n!)\ln p\approx n$ (leading order), so the total "weight" of primes up to $n$ contributes about $n$ in the log.

Conclusion. Stirling is compatible with the prime number theorem: $\sum_{p\le n}\ln p\sim n$ (Chebyshev's $\theta$ function).