We shall describe the principles of various methods for computing π(N) for a given N. We leave it to readers to translate these principles into their chosen programming language or package.

The simplest method is the well-known “sieve of Eratosthenes”. It operates as follows to create a list of primes up to a chosen N, from which one can read off values of π(n). Start with the full list of integers from 2 to N. Delete all multiples of 2, apart from 2 itself. Then delete all multiples of 3 (apart from 3 itself). Since 4 has already been deleted, move on to 5 and delete its multiples. Continue the process, deleting multiples of each m < N½ unless m has already been deleted. Note that when deleting multiples of m, it is enough to start at m2, since the numbers km (for 2 ≤ k < m) have already been deleted. When we have finished, the remaining numbers are the primes in [2, N], since every composite number in this interval is a multiple of some m ≤ N½, and consequently has been deleted.

To implement this process on a computer, what we really do is construct the function

(in the notation of this book, f is up). We start by setting f(n) = 1 for 2 ≤ n ≤ N. The act of “deleting” an integer n equates to changing f(n) from 1 to 0.