One recent proof of quadratic reciprocity involves computing various rations of the Gauss sum. 




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In [**Quadratic reciprocity and the sign of the Gauss sum via the finite Weil representation**](http://arxiv.org/abs/0808.2447) Gurevich, Hadani and Howe discuss the [Gauss sums](https://en.wikipedia.org/wiki/Gauss_sum), which are finite field analogues of the Gamma function $\Gamma(n+1)$:


$$ G\_p = \sum\_{x \in \mathbb{F}\_p} e^{\frac{2\pi i x^2}{p}} = 
\left\{ \begin{array}{rc} \sqrt{p} & \text{if }p \equiv 1 \mod 4 \\ 
i\sqrt{p} & \text{if }p \equiv 3 \mod 4\end{array}\right.$$


with the relationship $G\_p^2 = (\tfrac{-1}{p})\cdot p$. Using properties of the [Weil representation](http://arxiv.org/abs/1108.0202) of $SL(2,\mathbb{Z})$, and the fact that the Finite Fourier Transform corresponds to


$$ S = \left( \begin{array}{cr} 0 & -1 \\ 1 & 0 \end{array} \right)$$


they are able to prove quadratic reciprocity and correctly get the sign of the Legendre symbol. In fact, they compute the ratio:


$$ (\tfrac{p}{q}) (\tfrac{q}{p}) = \frac{G\_p \cdot G\_q}{G\_{pq}}$$




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The expression on the right looks like a [Mobius function](https://en.wikipedia.org/wiki/M%C3%B6bius_function) for the divisors of $pq$. So I started trying other combinations to see if I got meaningful symbols. Using three variables:


$$ \frac{G\_p G\_q G\_r G\_{pqr}}{G\_{pq}G\_{qr}G\_{pr}} \equiv 1$$


Computer experiments showed this ratio always to be $1$. In order to fine more interesting behavior I found:


$$ \frac{G\_p G\_q G\_r }{G\_{pqr}} \equiv \pm 1$$


And there is no reason to stop there since Mobius functions can be defined for any [lattice](http://www.math.cornell.edu/~levine/18.312/alg-comb-lecture-12.pdf) using [incidence algebra](https://en.wikipedia.org/wiki/Incidence_algebra). In particular, the poset of numbers and their divisors. 


Is there a name for these generalized symbols, or are these trivial extensions of the original Legendre symbol? Finally, what is the rule determining the sign in the second example?



higher reciprocity theorems from ratios of Gauss sums
If you look at bit further in the preprint you will see that the authors also state the generalisation
\begin{equation\*}
\frac{G\_{n\_1} G\_{n\_2}}{G\_{n\_1 n\_2}} = \left(\frac{n\_1}{n\_2}\right) \left(\frac{n\_2}{n\_1}\right)
\end{equation\*}
for any coprime odd numbers $n\_1$, $n\_2$. From this it easy to see that indeed
\begin{equation\*}
\frac{G\_p G\_q G\_r G\_{pqr}}{G\_{pq} G\_{qr} G\_{pr}}=1,
\end{equation\*}
and that
\begin{equation\*}
\frac{G\_p G\_q G\_r}{G\_{pqr}}=
\begin{cases}
+1 & \mbox{if at most one of $p$, $q$ and $r$ is $\equiv 3\pmod{4}$,}\\
-1 & \mbox{otherwise.}
\end{cases}
\end{equation\*}
In general the formula also shows that such "balanced" quotients as above always can be converted to a symmetric product of Legendre symbols which in turn only depends on the congruence classes mod $4$ for the primes involved. 


