This chapter concerns itself with the question: how many primes are there? In Chapter 1, we proved that there are infinitely many primes; however, we are interested in a more quantitative answer to this question; that is, we want to know how “dense” the prime numbers are.

This chapter has a bit more of an “analytical” flavor than other chapters in this text. However, we shall not make use of any mathematics beyond that of elementary calculus.

Chebyshev's theorem on the density of primes

The natural way of measuring the density of primes is to count the number of primes up to a bound x, where x is a real number. For a real number x ≥ 0, the function π(x) is defined to be the number of primes up to x. Thus, π(1) = 0, π(2) = 1, π(7.5) = 4, and so on. The function π is an example of a “step function,” that is, a function that changes values only at a discrete set of points. It might seem more natural to define π only on the integers, but it is the tradition to define it over the real numbers (and there are some technical benefits in doing so).

Let us first take a look at some values of π(x). Table 5.1 shows values of π(x) for x = 103i and i = 1, …, 6.