In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number x. [1][2] It is denoted by π(x) (unrelated to the number π). A symmetric variant seen sometimes is π0(x), which is equal to π(x) − 1⁄2 if x is exactly a prime number, and equal to π(x) otherwise.

The prime counting function is the function pi(x) giving the number of primes less than or equal to a given number x (Shanks 1993, p. 15). For example, there are no primes <=1, so pi(1)=0. There is a single prime (2) <=2, so pi(2)=1.

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In mathematics, at least four different functions are known as the pi or Pi function: π ( x ) {\displaystyle \pi (x)\,\!} ( pi function ) – the prime-counting function Π ( x ) {\displaystyle \Pi (x)\,\!}

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The following table contains values of pi(x), the number of prime numbers less than x : p (x) = # { p £ x, p is prime}. For example, p (11) = 5 since there are 5 primes less than or equal to 11, which are 2, 3, 5, 7 and 11.

The Greek letter π (pi) is a geometric constant approximately equal to 3.1416. Its value is equal to the length of any circle's circumference divided by its diameter.

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