On Bilinear Exponential and Character Sums with Reciprocals of Polynomials

We give nontrivial bounds for the bilinear sums $$\sum _{u=1}^U\sum _{v=1}^V{\alpha }_u{\beta }_v{\mathbf{e}}_p(u\mathrm{/}f(v))$$ where ${\mathbf{e}}_p(z)$ is a nontrivial 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 $\{{\alpha }_u\}$, $\{{\beta }_v\}$. 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 ${\mathbb{F}}_p$.