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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.

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Which Gauss sum? You are not giving much information. – Igor Rivin Dec 20 '11 at 12:17

The connection between Gauss sums and the $p$-adic Gamma function is a very deep theorem, and given by the Gross-Koblitz formula.

Here is the original paper by Gross and Koblitz: Gauss sums and the $p$-adic $\Gamma$-function. (1979)

For some more specific cases and applications, see the book by Berndt, Evans, and Williams, "Gauss and Jacobi Sums. (1998)"

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See Very lucid and freely available. (Helversen-Pasotto, A.
Gamma-function and Gaussian-sum-function.)

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Whoever down voted this: why? – Igor Rivin Dec 20 '11 at 15:32

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.

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