An estimate for the sum of Legendre symbols
Let \(p\) be an odd prime, \( \mathbb{F}_{p}[x]\) the ring of polynomials over the prime finite field \( \mathbb{F}_{p}\), and \(f(x) \in \mathbb{F}_{p}[x]\) a polynomial of odd degree \(n \geq 3\). If \[ S(f)= \sum_{x=0}^{p-1} \left( \frac{f(x)}{p} \right) \] is the sum of the Legendre symbols of \(f(x)\) then the well-known result of \textit{A. Weil} [Sur les courbes algébriques et les variétés qui s'en déduisent. Actualités Sci. Ind., No. 1041 = Publ. Inst. Math. Univ. Strasbourg 7 (1945). Paris: Hermann (1948; Zbl 0036.16001)] implies \[ |S(f)| \leq (n-1) \sqrt{p}. \] In 1969 an elementary proof of this inequality was given by the reviewer in [Izv. Akad. Nauk SSSR, Ser. Mat. 33, 1171--1181 (1969; Zbl 0192.58002)]. Then using a variant of the reviewer's construction \textit{N. M. Korobov} [Sov. Math., Dokl. 12, 241--245 (1971); translation from Dokl. Akad. Nauk SSSR 196, 764--767 (1971; Zbl 0233.10019)] proved that \[ |S(f)| \leq (n-1) \sqrt{p- \frac{(n-3)(n-4)}{4}} \quad \text{for} \quad p \geq \frac{n^{2}+9}{2}. \] The author slightly refines Korobov's construction and obtains the following somewhat stronger result \[ |S(f)| < (n-1) \sqrt{p- \frac{(n-3)(n+1)}{4}} \quad \text{for} \quad p \geq \frac{(n^{2}-1)}{2}. \]
Sum of Legendre symbols with a polynomial
scientific article; zbMATH DE number 5980726