Algebraic number theory LTCC 2009

10. Dirichlet characters and Dirichlet L-functions Definition 10.1. Let m ∈ N. A Dirichlet character modulo m is a function χ : N→ C that satisfies the following three conditions. (1) χ is periodic with period m, i.e. if a ≡ b (mod m) then χ(a) = χ(b). (2) χ is completely multiplicative, i.e. χ(ab) = χ(a)χ(b) for all a, b ∈ N. (3) χ(a) = 0 if and only if gcd(a,m) > 1. For m ∈ N we let (Z/mZ)× denote the multiplicative group of the ring Z/mZ. By a character of (Z/mZ)× we mean a homomorphism ψ : (Z/mZ)× → C×. Proposition 10.2. Let m ∈ N. If ψ : (Z/mZ)× → C× is a character of (Z/mZ)×, then the function χ : N→ C defined by