Gaussian Sum

$$S(p,q)\equiv \sum_{r=0}^{q-1} e^{-\pi ir^2p/q},$$ (1)

where $p$ and $q$ are Relatively Prime Integers. If $(n,n')=1$, then

$$S(m,nn')=S(mn',n)S(mn,n').$$ (2)

Gauß showed

$$\sum_{r=0}^{q-1} e^{2\pi i r^2/q} = {1-i^q\over 1-i} \sqrt{q}$$ (3)

for Odd $q$. A more general result was obtained by Schaar. For $p$ and $q$ of opposite Parity (i.e., one is Even and the other is Odd), Schaar's Identity states

$${1\over \sqrt{q}} \sum_{r=0}^{q-1} e^{-\pi ir^2p/q} = {e^{-\pi i/4}\over \sqrt{p}} \sum_{r=0}^{p-1} e^{\pi ir^2q/p}.$$ (4)

Such sums are important in the theory of Quadratic Residues.

See also Kloosterman's Sum, Schaar's Identity, Singular Series

References

Evans, R. and Berndt, B. ``The Determination of Gauss Sums.'' Bull. Amer. Math. Soc. 5, 107-129, 1981.

Katz, N. M. Gauss Sums, Kloosterman Sums, and Monodromy Groups. Princeton, NJ: Princeton University Press, 1987.

Riesel, H. Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkhäuser, pp. 132-134, 1994.