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

In **Quadratic reciprocity and the sign of the Gauss sum via the finite Weil representation** Gurevich, Hadani and Howe discuss the Gauss sums, 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 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:

$$ (\tfrac{p}{q}) (\tfrac{q}{p}) = \frac{G_p \cdot G_q}{G_{pq}}$$

The expression on the right looks like a Mobius 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 using 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?