TITLE: Is $\ p_n^{\pi(n)} < 4^n$ where $p_n$ is the largest prime $\leq n$?
QUESTION [9 upvotes]: Is $\ p_n^{\pi(n)} < 4^n$ where $p_n$ is the largest prime $\leq n$?
Where $\pi(n)$ is the prime counting function. Using PMT it seems asymptotically $\ p_n^{\pi(n)} \leq x^n$ where $e \leq x$

REPLY [7 votes]: Using $$\pi(x) \le 1.25066 \frac{x}{\log x}$$ for all $x>1$ (from Rosser and Schoenfeld), you have
$$(p_n)^{\pi(n)} \le e^{1.25066 n}  < 3.5^n$$ for all $n\ge 2$.