Primitive Dirichlet Characters

Let \chi be a Dirichlet characters of modulus q. Thus \chi is a periodic function of period q. It is possible, that for values of a restricted by condition \gcd(a,q)=1, the function \chi may have d period less than q.

Definition (Primitive characthers) We say that a Dirichlet character \chi  of modulus q is imprimitive if \chi(a) restricted by \gcd(a,q)=1 has a period which is less than q, Otherwise we say that \chi is primitive.

For maximal clarity, let us write out exactly what we mean \chi(a) restricted (a,q)=1 has a period n\in\mathbf{Z}^+. This means: For any integers a,b with \gcd(a,q)=\gcd(b,q)=1 and a=b\ \text{mod}\ n we have \chi(a)=\chi(b).

Definition (Conductor) Given a Dirichlet character \chi of modulus q, the conductor of \chi is defined to be the smallest positive integer q' such that \chi(a) restricted by \gcd(a,q)=1 has a period q'.

Lemma If \chi is a Dirichlet character of modulus q and if q_1\in\mathbf{Z}^+ is a period of \chi(a) restricted by \gcd(a,q)=1, then c(\chi)\mid q_1. Hence in particular ,we have c(\chi)\mid q.

Lemma For each Dirichlet character \chi of modulus q, there is a unique Dirichlet character of \chi' of modulus c(\chi) such that

\displaystyle \chi(n)=\begin{cases}\chi'(n) & \text{if}\ \gcd(a,q)=1\\  0 & \text{if} \gcd(a,q)>1.  \end{cases}

This Dirichlet character \chi' is primitive

 

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