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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. –  quanta Mar 17 '11 at 1:08
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NIST Digital Library of Mathematical Functions has a section on Periodic Number-Theoretic Functions that might be helpful. –  Eric Nitardy Mar 18 '11 at 2:55
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