Random Sieving and the Prime Number Theorem

Abstract

The Prime Number Theorem is a remarkable and rather deep result in the theory of numbers. The aim of this article is to show that this theorem can be made plausible by quite simple and elementary methods which, in addition, give a fascinating insight into the stochastic nature of that theorem. The material of this article is intended for use at the college level to integrate and motivate chapters on probability, sequence and series (especially the harmonic series), and the logarithmic function usually treated in most elementary mathematics courses. The heuristic arguments in this article can be tightened up, but that cannot be done at college level. For every real number x let 7r(x) be the number of primes less than or equal to x. One finds that 7r(10) = 4, 7r(100) = 25, 7r(1000) = 168, etc. The function x -* 7r(x) will be called the prime number function. All attempts to find a formula for 7r(x) representing 7r(x) in "closed form" by a finite number of "known" functions have failed, and will necessarily fail. There are, however, some simple asymptotic expressions for 7r(x), such as