Prime-counting function

Function representing the number of primes less than or equal to a given number The values of π(n) for the first 60 positive integers In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number x. 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. That is, the number of prime numbers less than x, plus half if x equals a prime. Growth rate Of great interest in number theory is the growth rate of the prime-counting function. It was conjectured in the end of the 18th century by Gauss and by Legendre to be approximately x log ⁡ x {\displaystyle {\frac {x}{\log x}}} where log is the natural logarithm, in the sense that lim x → ∞ π ( x ) x / log ⁡ x = 1. {\displaystyle \lim _{x ightarrow \infty }{\frac {\pi (x)}{x/\log x}}=1.} This statement is the prime number theorem. An equivalent statement is lim x → ∞ π ( x ) li ⁡ ( x ) = 1 {\displaystyle \lim _{x ightarrow \infty }{\frac {\pi (x)}{\operatorname {li} (x)}}=1} where li is the logarithmic integral func