Precursors

Much of analytic number theory was inspired by the prime number theorem. Let ?(x) be the prime-counting function that gives the number of primes less than or equal to x, for any real number x. For example, ?(10) = 4 because there are four prime numbers (2, 3, 5 and 7) less than or equal to 10. The prime number theorem then states that x / ln(x) is a good approximation to ?(x), in the sense that the limit