This article is cited in 2 papers

Estimate of a sum of Legendre symbols of polynomials of even degree

D. A. Mit'kin

M. V. Lomonosov Moscow State University

Abstract: Let $n\ge4$ be even, $p>\frac{n^2-2n}2$ be simple odd, and $f(x)=a_0+a_1x+\dots+a_nx^n$ be a polynomial with integral coefficients that are not quadratic over the residue field modulo $p$, $(a_n,p)=1$. The following inequality is proved:
$$ \biggl|\sum_{x=1}^p\biggl(\frac{f(x)}p\biggr)\biggr|\le(n-2)\sqrt{p+1-\frac{n(n-4)}4}+1. $$


UDC: 511.3

Received: 07.07.1972

 English version: Mathematical Notes, 1973, 14:1, 597–602

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