Geodesic
On a New Exponential Sum
Canadian mathematical bulletin, Tome 44 (2001) no. 1, pp. 87-92

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DOI

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 $\varepsilon \,>\,0$ $$\underset{\gcd (a,\,p)\,=\,1}{\mathop{\max }}\,\,\left| S\left( a \right) \right|\,=\,O\left( {{t}^{5/6+\varepsilon }}\,{{p}^{1/8}} \right)$$
DOI : 10.4153/CMB-2001-010-1
Mots-clés : 11L07, 11T23, 11B50, 11K31, 11K38 
Lieman, Daniel; Shparlinski, Igor. On a New Exponential Sum. Canadian mathematical bulletin, Tome 44 (2001) no. 1, pp. 87-92. doi: 10.4153/CMB-2001-010-1
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     title = {On a {New} {Exponential} {Sum}},
     journal = {Canadian mathematical bulletin},
     pages = {87--92},
     year = {2001},
     volume = {44},
     number = {1},
     doi = {10.4153/CMB-2001-010-1},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2001-010-1/}
}
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