ON BILINEAR EXPONENTIAL AND CHARACTER SUMS WITH RECIPROCALS OF POLYNOMIALS

We give non-trivial bounds for the bilinear sums Sigma(U)(u=1)Sigma(V)(v=1) alpha(u)beta(v)e(p)(u/f(v)), where e(p)(z) is a non-trivial additive character of the prime finite field F-p of p elements, with integers U, V, a polynomial f is an element of 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 F-p.