= Finite Fourier transform of a primitive Dirichlet character
{title2=$S_\chi(r)=\overline\chi(r)g(\chi)$}
For a <primitive Dirichlet character>, $S_\chi(r)=\overline\chi(r)g(\chi)$ for every integer $r$, with zero on nonunits. For a unit $r$ this follows by substitution. If a prime $p$ divides both $r$ and $N$, primitivity supplies a unit $u\equiv1\pmod{N/p}$ with $\chi(u)\ne1$; substitution by $u$ fixes the additive exponential and forces $S_\chi(r)=0$. For an imprimitive <Dirichlet character> this vanishing can fail.
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