# derivation of Gauss sum up to a sign

The Gauss sum can be easily evaluated up to a sign by squaring the original series

 $\displaystyle g_{1}^{2}(\chi)$ $\displaystyle=\sum_{s\in\mathbb{Z}/p\mathbb{Z}}\left(\frac{s}{p}\right)e^{2\pi is% /p}\sum_{t\in\mathbb{Z}/p\mathbb{Z}}\left(\frac{t}{p}\right)e^{2\pi it/p}$ $\displaystyle=\sum_{s,t\in\mathbb{Z}/p\mathbb{Z}}\left(\frac{st}{p}\right)e^{2% \pi i(s+t)/p}$ and summing over a new variable $n=s^{-1}t\pmod{p}$ $\displaystyle=\sum_{s,n\in\mathbb{Z}/p\mathbb{Z}}\left(\frac{n}{p}\right)e^{2% \pi i(s+ns)}$ $\displaystyle=\sum_{n\in\mathbb{Z}/p\mathbb{Z}}\left(\frac{n}{p}\right)\sum_{s% \in\mathbb{Z}/p\mathbb{Z}}e^{2\pi is(n+1)}$ $\displaystyle=\sum_{n\in\mathbb{Z}/p\mathbb{Z}}\left(\frac{n}{p}\right)(q[n% \equiv-1\pmod{p}]-1)$ $\displaystyle=p\left(\frac{-1}{p}\right)-\sum_{n\in\mathbb{Z}/p\mathbb{Z}}% \left(\frac{n}{p}\right)$ $\displaystyle=p\left(\frac{-1}{p}\right)=\begin{cases}\hphantom{+{}}p,&{\rm if% \ }p\equiv 1\pmod{4},\\ -p,&{\rm if\ }p\equiv 3\pmod{4}.\end{cases}$

## References

• 1 Harold Davenport. Markham Pub. Co., 1967. http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0159.06303Zbl 0159.06303.
Title derivation of Gauss sum up to a sign DerivationOfGaussSumUpToASign 2013-03-22 13:39:45 2013-03-22 13:39:45 bbukh (348) bbukh (348) 8 bbukh (348) Derivation msc 11L05