Let $\chi : \mathbb Z \to \mathbb C$ be a Dirichlet character mod $k$ ; i.e. $\chi (m+k)=\chi (m) , \forall m \in \mathbb Z$ ; $\chi (mn)=\chi(m)\chi (n),\forall m,n \in \mathbb Z$ and $\chi (n)=0$ iff gcd $(m,k)\ne 1$ .
Let us call $r\in \mathbb Z$ a "period" of $\chi$ if $\chi(m+r)=\chi (m) , \forall m\in \mathbb Z$ such that gcd $(m,k)=1$ .
Then does the set of all periods of $\chi$ form a subgroup of $\mathbb Z$ ?
I can show $0$ is a period but I can't show the sum of periods is period or even the negative of a period a period , mainly due to the restriction that the equality is satisfied for gcd $(m,k)=1$.
Please help. Thanks in advance