I was wondering if there exists a Poisson Summation formula (like the one existing with primitive character) for imprimitive Dirichlet characters ?


For a primitive Dirichlet character $\chi$ we have:


$$ \sum\limits\_{n=-\infty}^{\infty}\chi(n) f\bigg(\frac{n}{q}x\bigg) =\frac{K}{x} \sum\limits\_{n=-\infty}^{\infty} \overline{\chi(n)} \hat{f}\bigg(\frac{n}{x}\bigg)$$


But for imprimitive characters ?



Is there a Poisson Summation formula for imprimitive Dirichlet characters?
The reason we can get that (twisted) Poisson summation formula in the first place is that in the primitive case you can interpolate the character to a smooth real function via Gauss sums.


In the imprimitive case this is not the case anymore, and you can't get a function nice enough to anything that resembles a Poisson formula to hold.


Of course nothing is lost, since imprimitive characters are induced by primitive ones, and for example (since you have used the Dirichlet series tag), we have:


$$L(\chi,s)=\prod\_{\substack{
 p|m \\
 p\nmid f
 }}
 (1-\chi (p)p^{-s})L(\chi ',s)$$


which gives you functional equation and analytic continuation of Dirichlet series for imprimitive $\chi'$. This kind of induced-character argument bypasses the need for Poisson summation in any case I can think of.


