Notes on Primes in Arithmetic Progression

The following is a quick set of notes of some properties of Dirichlet characters, in particular, how they are used to prove the infinitude of primes in arithmetic progressions. These notes are from from An Invitation to Modern Number Theory, by myself and Ramin Takloo-Bighash. As this is a modified snippet from the book, references to other parts of the book are displayed as ??. 1. Dirichlet Characters 1.1. Dirichlet Characters. Let m be a positive integer. A completely multiplicative (see Definition ??) arithmetic function with periodm that is not identically zero is called a Dirichlet character. In other words, we have a function f : Z → C such that f(xy) = f(x)f(y) and f(x +m) = f(x) for all integers x, y. Often we call the period m the conductor or modulus of the character. Exercise 1.1. Let χ be a Dirichlet character with conductor m. Prove χ(1) = 1. If χ is not identically 1, prove χ(0) = 0. Because of the above exercise, we adopt the convention that a Dirichlet character has χ(0) = 0. Otherwise, given any character, there is another character which differs only at 0. A complex number z is a root of unity if there is some positive integer n such that z = 1. For example, numbers of the form e are roots of unity; if a is relatively prime to q, the smallest n that works is q, and we often say it is a q root of unity. Let χ0(n) = { 1 if (n,m) = 1 0 otherwise. (1) We call χ0 the trivial or principal character (with conductor m); the remaining characters with conductor m are called non-trivial or non-principal. Exercise 1.2. Let χ be a non-trivial Dirichlet character with conductor m. Prove that if (n,m) = 1 then χ(n) is a root of unity, and if (n,m) 6= 1 then χ(n) = 0. Theorem 1.3. The number of Dirichlet characters with conductor m is φ(m). Proof. We prove the theorem for the special case of m prime. By Theorem ?? the group (Z/pZ)∗ is cyclic, generated by some g of order p− 1. Thus any x ∈ (Z/pZ)∗ is equivalent to g for some k depending on x. As χ(g) = χ(g), once we have determined the Dirichlet character at a generator, its values are determined at all elements (of course, χ(0) = χ(m) = 0). By Exercise 1.2, χ(g) is a root of unity. As gp−1 ≡ 1 mod p and χ(1) = 1, χ(g)p−1 = 1. Therefore χ(g) = e2πia/(p−1), a ∈ {1, 2, . . . , p − 1}. The proof is completed by noting each of these possible choices of a gives rise to a Dirichlet character, and all the characters are distinct (they have different values at g). Not only have we proved (in the case of m prime) how many characters there are, but we have a recipe for them. If a = p− 1 in the above proof, we have the trivial character χ0. Exercise 1.4 (Important). Let r and m be relatively prime. Prove that if n ranges over all elements of Z/mZ then so does rn (except in a different order if r 6≡ 1 mod m). i