TITLE: Gamma-function analogues for Gauss sums
QUESTION [7 upvotes]: I have a Gauss sum, which I have to calculate. I have heard that it has an analogues form with the Gamma function, but couldn't find its formula shape. It would be so nice of you to help me and write the mathematical shape of the link between these two functions. Beforehand thank you.

REPLY [6 votes]: This is a great question. Suppose $K$ is a locally compact second countable field, and $K^*$ the multiplicitative group of nonzero elements of $K$. Suppose $\phi$ is a homomorphism of the additive group $K$ into $S^1$ and suppose $\chi $ is a homomorphism of the multiplicative group  $K^*$ into $S^1$. If the integral 
$$\int _{K^*} \phi (x) \chi (x)d^*x $$ makes sense (even as a distribution), then this can be called the "Gamma Function" of $K$.
If we take $K={\mathbb Z}/p{\mathbb Z}$ for a prime $p$, and $\phi(x)=e^{2\pi i x/p}$ , and $\chi (x)$ a character on $K^*$ then we get a Gauss sum. 
If we take $K={\mathbb R}$, $\phi (x)=e^{x}$, and $\chi (x)= \mid x \mid ^s$then the above "integral" is the Gamma function. Here the integral , if interpreted as a functional on compactly supported functions on positive reals, makes sense.
In this sense, the Gamma function, or the Gauss sum, is the Mellin transform of an additive character of $K$ with respect to a multiplicative char of the group $K^*$ of non-zero elements.