Abstract:
For the sum $S$ of the Legendre symbols of a polynomial of odd degree $n\ge3$ modulo primes $p\ge3$, Weil's estimate $|S|\le(n-1)\sqrt p$ and Korobov's estimate
$$
|S|\le (n-1)\sqrt{p-\frac{(n-3)(n-4)}{4}}\qquad \text{for}\quad p\ge\frac{n^2+9}{2}
$$
are well known. In this paper, we prove a stronger estimate, namely,
$$
|S|<(n-1)\sqrt{p-\frac{(n-3)(n+1)}{4}}.
$$
Keywords:polynomial of odd degree, Legendre symbol, Weil's estimate, Korobov's estimate.
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