I have the following problem. Let $p$ be some prime. What is the value of \begin{equation} \sum_{k=1}^{p-1} \left(\frac{k+1}{p}\right) \omega_p^{kl}, \end{equation} where $\left(\frac{k+1}{p}\right)$ is the Legendre symbol, and $\omega_p = e^{\frac{2\pi i}{p}}.$ [solved].

But what is the value of \begin{equation} \sum_{k=1}^{p-1} \left(\frac{k^2+k}{p}\right) \omega_p^{kl}? \end{equation}

I found the standard result for $\left(\frac{k}{p}\right)$, $\sqrt{p}$ or $i\sqrt{p},$ but I don't know the proof techniques and therefore don't know how to approach this one. Any ideas? I am not specialist in number theory, and I don't even know if it is easy or hard question :)

Any hints or links to references are welcomed.

What I actually need is the value (or a lower bound of the absolute value) of a Gauss sum with $\chi(k) = (\left(\frac{k}{p}\right)+1)(\left(\frac{k+1}{p}\right)+1).$

quadraticso the exponent doesn't matter: $a^{-1} = a$ if $a = \pm 1$. Concerning a sum with $(\frac{k^2+k}{p}) = (\frac{k(k+1)}{p})$ in it, replacing $k$ with $-k$ makes that $(\frac{-1}{p})(\frac{k(1-k)}{p})$ and then you basically have a Jacobi sum, which you can look up elsewhere. This is not really a research-level question. I suggest if you have similar questions that you ask them on math.stackexchange. $\endgroup$ – KConrad May 16 '15 at 21:0913more comments