Character sums over smooth numbers

Abstract

Let $\hat{I}¨(x,y)$ denote the count of $y$-smooth numbers below $x$ and $P(n)$ denote the largest prime factor of $n$. We show that \[ \frac{1}{φ(q)} \sum_{χ\bmod q} \Bigl| \sum_{\substack{n \leq x \\ P(n) \leq y}} χ(n) \Bigr| = o \Bigl( \sqrt{Ψ(x,y)} \Bigr), \] whenever $(\log x{)}^6\le y\le x^{\frac{1}{32\log \log x}}$ and $q\ge x^{1+\epsilon }$ for some small quantifiable $\epsilon >0$. The saving is substantial when $\epsilon$ is fixed away from zero, and we prove similar results for continuous characters and completely multiplicative twists of these sums.

16 pages