Prime-counting function

The prime-counting function $\pi (X)$ is the number of primes at most $X$.

For $a=\pi (\sqrt{x})$, let $p_1,\dots ,p_a$ be the primes at most $\sqrt{x}$. The inclusion-exclusion principle counts unsieved positive integers by $\varphi (x,a)={\sum }_S(-1{)}^{|S|}\lfloor x/{\prod }_{i\in S}{p}_i\rfloor$. These are $1$ and the remaining primes, since every composite number has a prime factor at most its square root. Hence $\pi (x)=\varphi (x,a)+a-1$ for $x\ge 2$. This is a prime-counting identity, distinct from the Legendre formula for factorial valuations.

If $P(X)\le A^X$ for every $X\ge 1$, where $A>1$, thenfor $X\ge 2$. Every prime in $(\sqrt{X},X]$ contributes at least $\sqrt{X}$ to the primorial, soTaking logarithms, using $\pi (\sqrt{X})\le \sqrt{X}$, and observing that $\log X\le \sqrt{X}$ for $X\ge 2$ proves the claim.