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On a New Exponential Sum

Published online by Cambridge University Press:  20 November 2018

Daniel Lieman  and
Igor Shparlinski
Daniel Lieman
Affiliation:
Department of Mathematics University of Missouri Columbia, Missouri 65211 USA, email: [email protected]
Igor Shparlinski
Affiliation:
Department of Computing Macquarie University NSW 2109 Australia, email: [email protected]
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Abstract

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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 $\varepsilon \,>\,0$

$$\underset{\gcd (a,\,p)\,=\,1}{\mathop{\max }}\,\,\left| S\left( a \right) \right|\,=\,O\left( {{t}^{5/6+\varepsilon }}\,{{p}^{1/8}} \right)$$

Keywords

11L0711T2311B5011K3111K38
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 44 , Issue 1 , 01 March 2001 , pp. 87 - 92
Copyright
Copyright © Canadian Mathematical Society 2001

References

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