Exponential sums with reducible polynomials

Abstract

Hooley proved that if $f\in \mathbb{Z}[X]$ is irreducible of degree $\ge 2$, then the fractions $\{r/n\}$, $0<r<n$ with $f(r)\equiv 0(\mathrm{mod}n)$, are uniformly distributed in $(0,1)$. In this paper we study such problems for reducible polynomials of degree $2$ and $3$ and for finite products of linear factors. In particular, we establish asymptotic formulas for exponential sums over these normalized roots.

31 pages