Recently, I am considering a class of character sums concerning Legendre symbols. Let $p$ be an odd prime, and $\phi$ the Legendre symbol mod $p$. It is well known that $$\sum_x\phi(x+a)\phi(x+b)=-1,\ \ (a-b,p)=1,$$where $x$ runs over the residue system mod $p$. However, I haven't found any references on the character sums $$\sum_x\phi(x+a)\phi(x+b)\phi(x+c)$$ and $$\sum_x\phi(x+a)\phi(x+b)\phi(x+c)\phi(x+d).$$

Of courese, the sums of such types are useful as studying the consecutive quadratic residues.

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    $\begingroup$ arithboy: If some day you want to become arithman, you need to find a better notation for Legendre symbols than the letter phi. $\endgroup$ – KConrad Nov 2 '10 at 4:11
  • $\begingroup$ In Dieudonne's article "On the history of the Weil conjectures" he discusses the problem of counting consecutive squares mod p and uses sums like the ones in your question. The article originally appeared in The Mathematical Intelligencer 10, 1975 and it was reprinted as an appendix to Freitag and Kiehl's "Etale cohomology and the Weil conjecture". $\endgroup$ – KConrad Nov 2 '10 at 4:18
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    $\begingroup$ I just follow the notations of K. Ono and R. Evans. See "Values of Gaussian hypergeometric series" for instance. Neverthless, thanks for your reply and help. $\endgroup$ – arithboy Nov 2 '10 at 10:22

The sums that you give are related to the number of points on

$y^2 = (x+a)(x+b)(x+c)$ and $y^2=(x+a)(x+b)(x+c)(x+d)$ respectively.

Assuming that $a,b,c,d$ are distinct, the first is an elliptic curve, and the second a curve of genus 2 also a curve of genus 1. By the "Riemann Hypothesis" for curves over finite fields each has the value

$a_p$ where $a_p$ is the trace of Frobenius. In the both cases first case $|a_p| \le 2 \sqrt{p}$ and in the second $|a_p| \le 2 \lfloor 2 \sqrt{p} \rfloor$ (a result of Serre strengthening the normal RH) .

[added: the formula I gave includes the "point at infinity" so that you need to subtract 1 in the first case and 2 in the second. The first case that you give corresponds to a curve of genus 0].


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