Prime Number Theorem

The prime number theorem describes the asymptotic distribution of the prime numbers among the positive integers. It states that the number of prime numbers less than or equal to a given number \( x \) is approximately \[ \pi(x) \sim \frac{x}{\log x} \] where \( \pi(x) \) is the prime-counting function and \( \log x \) is the natural logarithm of \( x \).