Geodesic
An Estimate for the Sum of Legendre Symbols
Matematičeskie zametki, Tome 88 (2010) no. 6, pp. 859-866

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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, Weil's estimate, Korobov's estimate.
Mots-clés : Legendre symbol 
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     author = {E. A. Grechnikov},
     title = {An {Estimate} for the {Sum} of {Legendre} {Symbols}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {859--866},
     publisher = {mathdoc},
     volume = {88},
     number = {6},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2010_88_6_a5/}
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E. A. Grechnikov. An Estimate for the Sum of Legendre Symbols. Matematičeskie zametki, Tome 88 (2010) no. 6, pp. 859-866. http://geodesic.mathdoc.fr/item/MZM_2010_88_6_a5/