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ON BILINEAR EXPONENTIAL AND CHARACTER SUMS WITH RECIPROCALS OF POLYNOMIALS

  • Igor E. Shparlinski (a1)

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 $\mathbb{F}_{p}$ .

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