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On Kloosterman Sums with Oscillating Coefficients
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- Journal:
- Canadian Mathematical Bulletin / Volume 42 / Issue 3 / 01 September 1999
- Published online by Cambridge University Press:
- 20 November 2018, pp. 285-290
- Print publication:
- 01 September 1999
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- Article
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In this paper we prove: for any positive integers
$a$ and 
$q$ with 
$\left( a,\,q \right)\,=\,1$ , we have uniformly
$$\sum\limits_{\begin{matrix} n\le N\\ (n,q)=1,n\bar{n}\equiv 1(\,\bmod \,q)\\\end{matrix}}{\mu (n)e(\frac{a\bar{n}}{q})\ll Nd(q)\left\{ \frac{{{\log }^{\frac{5}{2}}}N}{{{q}^{\frac{1}{2}}}}+\frac{{{q}^{\frac{1}{5}}}{{\log }^{\frac{13}{5}}}N}{{{N}^{\frac{1}{5}}}} \right\}.}$$This improves the previous bound obtained by D. Hajela, A. Pollington and B. Smith [5].
