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ON BILINEAR EXPONENTIAL AND CHARACTER SUMS WITH RECIPROCALS OF POLYNOMIALS

Part of: Exponential sums and character sums Diophantine equations

Published online by Cambridge University Press:  16 May 2016

Igor E. Shparlinski
Igor E. Shparlinski*
Affiliation:
Department of Pure Mathematics, University of New South Wales, Sydney, NSW 2052, Australia email igor.shparlinski@unsw.edu.au
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Abstract

We give non-trivial bounds for the bilinear sums

$$\begin{eqnarray}\mathop{\sum }_{u=1}^{U}\mathop{\sum }_{v=1}^{V}\unicode[STIX]{x1D6FC}_{u}\unicode[STIX]{x1D6FD}_{v}\,\mathbf{e}_{p}(u/f(v)),\end{eqnarray}$$
where $\,\mathbf{e}_{p}(z)$ is a non-trivial additive character of the prime finite field $\mathbb{F}_{p}$ of $p$ elements, with integers $U$, $V$, a polynomial $f\in \mathbb{F}_{p}[X]$ and some complex weights $\{\unicode[STIX]{x1D6FC}_{u}\}$, $\{\unicode[STIX]{x1D6FD}_{v}\}$. In particular, for $f(X)=aX+b$, we obtain new bounds of bilinear sums with Kloosterman fractions. We also obtain new bounds for similar sums with multiplicative characters of $\mathbb{F}_{p}$.

MSC classification

Primary: 11L07: Estimates on exponential sums
Secondary: 11D79: Congruences in many variables
Type
Research Article
Information
Mathematika , Volume 62 , Issue 3 , 2016 , pp. 842 - 859
Copyright
Copyright © University College London 2016 

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