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ON BILINEAR EXPONENTIAL AND CHARACTER SUMS WITH RECIPROCALS OF POLYNOMIALS
Published online by Cambridge University Press: 16 May 2016
Abstract
We give non-trivial bounds for the bilinear sums $$\begin{eqnarray}\mathop{\sum }_{u=1}^{U}\mathop{\sum }_{v=1}^{V}\unicode[STIX]{x1D6FC}_{u}\unicode[STIX]{x1D6FD}_{v}\,\mathbf{e}_{p}(u/f(v)),\end{eqnarray}$$
$\,\mathbf{e}_{p}(z)$ 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
- Type
- Research Article
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- Copyright
- Copyright © University College London 2016