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On a New Exponential Sum
Published online by Cambridge University Press: 20 November 2018
Abstract
Let $p$ be prime and let
$\vartheta \,\in \,\mathbb{Z}_{p}^{*}$ be of multiplicative order
$t$ modulo
$p$. We consider exponential sums of the form
$$S\left( a \right)\,=\,\sum\limits_{x=1}^{t}{\exp \left( 2\pi ia{{\vartheta }^{{{x}^{2}}}}\,/\,p \right)}$$
and prove that for any $\varepsilon \,>\,0$
$$\underset{\gcd (a,\,p)\,=\,1}{\mathop{\max }}\,\,\left| S\left( a \right) \right|\,=\,O\left( {{t}^{5/6+\varepsilon }}\,{{p}^{1/8}} \right)$$
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- Research Article
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- Copyright © Canadian Mathematical Society 2001
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