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In this chapter, we recall how natural numbers and integers may be constructively defined, and how to prove the basic rules of computation we learn in school. The purpose is to give a quick example of developing a mathematical theory from a set of very basic facts. The idea is to give the reader the capability of explaining to her/his grandmother why, for example, 2 times 3 is equal to 3 times 2. Answering questions of this nature leads to a deeper understanding of the nature of integers and the rules for computing with integers, which goes beyond just accepting facts you learn in school as something given once and for all. An important aspect of this process is the very questioning of established facts that follows from posing the why,which may lead to new insight and new truths replacing the old ones.

The method of complex integration together with the results of L-functions proved in Chapter VIII will enable us to write down an explicit formula connecting the sum of values of the function Λ(n) over the integers lying in a given arithmetic progression with the zeros of an L-function. This explicit formula together with a theorem on the boundary of the zeros of the L-function will yield the prime number theorem for arithmetic progressions. We shall always assume below that k ≤ x.

In this chapter we give some basic results concerning the distribution of prime numbers. The reader will only require some knowledge of the calculus —this chapter is a first introduction to analytic number theory and we shall omit all the deeper investigations.