Pi Function

The Pi function, which is written π(n), is the number of primes at or below n. The function is known exactly until π(10²³)= 201,467,286,689,315,906,290. The function is important in studying the Riemann Zeta Function, and also underlies much of the work on the Riemann Hypothesis. It also is related to other deep problems in modern number theory.

Several consequences of the value of this function are important to note. The first consequence is that the N-th prime is near where the approximation function puts π(n). The second consequence is that approximately π(n)/n numbers in the vicinity of n are prime, and therefore you can approximate how many numbers in ther area need to be tested to find a prime. This is important in cryptography, where primes of a certain size must be found.

An important branch of number theory deals with approximating π(n). The first real attempt at approximating the function was by Gauss, in 1791. He proved that π(n)~ n/ln(n). He later revised his approximation to π(n)~li(n), where li(n) is the logarithmic integral, given by ∫^x_1.451...(dt/ln(t)). This result is sometimes known as "the" prime number theorem. Since then, however, better approximations have been found by many people involved in number theory. Included in these are Legendre's Formula and Mape's Method. No function so far is a really great approximation of π(n), and the search continues.

Sources:
Lagarias, J. and Odlyzko, A. "Computing π(x): An Analytic Method."
Deleglise, M. and Rivat, J. "Computing π(x): The Meissel, Lehmer, Lagarias, Miller, Odlyzko Method."
Caldwell, C. K. "How Many Primes are there?" http://www.utm.edu/research/primes/howmany.shtml