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27 Functions of Number TheoryMultiplicative Number Theory

§27.8 Dirichlet Characters

If k (>1) is a given integer, then a function χ(n) is called a Dirichlet character (mod k) if it is completely multiplicative, periodic with period k, and vanishes when (n,k)>1. In other words, Dirichlet characters (mod k) satisfy the four conditions:

27.8.1 χ(1) =1,
27.8.2 χ(mn) =χ(m)χ(n),
m,n=1,2,,
27.8.3 χ(n+k) =χ(n),
n=1,2,,
27.8.4 χ(n) =0,
(n,k)>1.

An example is the principal character (mod k):

27.8.5 χ1(n)={1,(n,k)=1,0,(n,k)>1.

For any character χ(modk), χ(n)0 if and only if (n,k)=1, in which case the Euler–Fermat theorem (27.2.8) implies (χ(n))ϕ(k)=1. There are exactly ϕ(k) different characters (mod k), which can be labeled as χ1,,χϕ(k). If χ is a character (mod k), so is its complex conjugate χ¯. If (n,k)=1, then the characters satisfy the orthogonality relation

27.8.6 r=1ϕ(k)χr(m)χ¯r(n)={ϕ(k),mn(modk),0,otherwise.

A Dirichlet character χ(modk) is called primitive (mod k) if for every proper divisor d of k (that is, a divisor d<k), there exists an integer a1(modd), with (a,k)=1 and χ(a)1. If k is prime, then every nonprincipal character χ(modk) is primitive. A divisor d of k is called an induced modulus for χ if

27.8.7 χ(a)=1 for all a1 (mod d),
(a,k)=1.

Every Dirichlet character χ (mod k) is a product

27.8.8 χ(n)=χ0(n)χ1(n),

where χ0 is a character (mod d) for some induced modulus d for χ, and χ1 is the principal character (mod k). A character is real if all its values are real. If k is odd, then the real characters (mod k) are the principal character and the quadratic characters described in the next section.