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I usually work in number theory so I am not familiar with Fourier transforms, I have read up on them and know the basics but it never seems to be in number theory language.

I am trying to find the transform of a primitive Dirichlet character $\chi(n) \bmod q$. I know this is a periodic function and $\chi(n)=\exp\left(\frac{Kv(n)}{\phi(p^\alpha)}\right)$ but I have no idea have to find its transform or the transform of $f(n)\chi(n)$

Yes you are right, say how would you calculate $\sum_(n\epsilon Z) f(n)\chi(n)$

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    It should be the discrete Fourier transform? For example the discrete Fourier transform of the Legendre symbol function is related to the quadratic gauss sum.2011-03-17
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    *NIST Digital Library of Mathematical Functions* has a section on [Periodic Number-Theoretic Functions](http://dlmf.nist.gov/27.10) that might be helpful.2011-03-18

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