Character sums over shifted smooth numbers

Abstract

We give nontrivial bounds in various ranges for character sums of the form \[ ∑ n ∈ S ( x , y ) χ ( n + a ) , gcd ( a , p ) = 1 , \sum _{n\in \mathcal S(x,y)}\chi (n +a), \qquad \gcd (a,p) = 1, \] where χ \chi is a nontrivial multiplicative character modulo a prime p p and S ( x , y ) \mathcal S(x,y) is the set of positive integers n ≤ x n\le x that are divisible only by primes q ≤ y q \le y .