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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 of a Gauss sum with $\chi(k) = (\left(\frac{k}{p}\right)+1)(\left(\frac{k+1}{p}\right)+1).$

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 $\chi(k) = (\left(\frac{k}{p}\right)+1)(\left(\frac{k+1}{p}\right)+1).$

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 $\omega_p = e^{\frac{2\pi i}{p}}.$

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 of a Gauss sum with $\chi(k) = (\left(\frac{k}{p}\right)+1)(\left(\frac{k+1}{p}\right)+1).$

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 of a Gauss sum with $\chi(k) = (\left(\frac{k}{p}\right)+1)(\left(\frac{k+1}{p}\right)+1).$