 Prime-counting function

The prime-counting function $\pi (x)$ counts the prime numbers not exceeding a real bound $x$:Thus $\pi (10)=4$, $\pi (11)=5$ and $\pi (12)=5$. It is a nondecreasing step function with a jump of one at each prime number, and $\pi (x)=0$ for $x<2$.

Computing one large value is different from listing every prime number below it. A number-theoretic sieve can do both up to a moderate bound; Legendre prime-counting formula and the Lehmer prime-counting algorithm instead organize counts of allowable factors. The Prime number theorem describes the long-term scale $\pi (x)\sim x/\log x$, rather than exact individual jump locations.

The prime-counting function, denoted as \( \pi(x) \), is a mathematical function that counts the number of prime numbers less than or equal to a given number \( x \).