I've started to read the fifth chapter of "Multiplicative Number Theory" by Harold Davenport and I got stuck at some point. Let me elaborate the part that i didn't quite understand.
Let $\chi$ be any Dirichlet character to the modulus $q$ other than the principal character. It has been stated that if the function $\chi(n)$ may have a period less than $q$ for the values of $n$ restricted by the condition that $(n,q) = 1$, let us say the least period is $m$, clearly $m|q$.
But I didn't understand clearly what is being said here, because for the values of $n$ , $(n,m) =1 $ but $(n,q)>1$, we can find integer $t$ such that $(q,n+tm) = 1$, In other words $\chi(n) = 0$ but $\chi(n+tm) \neq 0$ because $(n+tm,q) =1$, however $\chi(n+tm) = \chi(n)$ .
I didn't understand that construction.