Prime-counting function

Definition

For a real number $x$, the prime-counting function $\pi(x)$ is the number of prime numbers $p$ with $p \le x$.

Remarks

For example, $\pi(10) = 4$, because the primes up to $10$ are $2, 3, 5, 7$. Also $\pi(100) = 25$. The function is a step function. It increases by $1$ at each prime. The prime number theorem says that $\pi(x)$ grows like $x / \ln x$. The symbol $\pi$ here has no relation to the number $3.14159\dots$.

Explanations

L1 human

The function $\pi(x)$ answers the question: how many primes are there up to $x$? The answer grows without bound, by Euclid's theorem, but the primes become rarer as $x$ grows.