Estimates for trilinear and quadrilinear character sums

Abstract

We obtain new bounds on some trilinear and quadrilinear character sums, which are non-trivial starting from very short ranges of the variables. An application to a modular analogue of a multiplicative hybrid problem of Iwaniec and S\'ark\"ozy (1987) is given. Moreover, we show that for all large primes $p$, there are primes $p_1, p_2, p_3$ of sizes $p^{{\frac 15}+\frac 1{1987}}$ such that $p_1p_2+p_3 $ is a quadratic non-residue modulo $p$, and there are also primes $q_1, q_2, q_3, q_4$ of sizes $p^{{\frac 18}+\frac 1{2024}}$ such that $q_1q_2+q_3 q_4 $ is a quadratic non-residue modulo $p$.

Locations

• arXiv (Cornell University) - View - PDF