Complexity $log(p_{n}\#)$

Question

Is $log(p_{n}\#)$ a polynomial complexity (we count complexity after $n$)?

$p_{n}\#$ is primorial.

Could anyone prove it (if possible)?

Answer 1

$\ln \prod \limits_{k=1}^{n} p_k = \sum \limits_{k=1}^{n} \ln p_k $

$\sum \limits_{k=1}^{n} \ln p_k = \theta(p_n) $ By definition (Read Chebyshev Theta function).

By the PNT $|\theta(p_n)-p_n| \leq \frac{p_n}{\ln p_n} $

And $ n \ln n < p_n <2 n \ln n$ using these one concludes that its $O(n \ln n)$ complexity.