A refinement of the Burgess bound for character sums

In this paper we give a refinement of the bound of D. A. Burgess for multiplicative character sums modulo a prime number $q$. This continues a series of previous logarithmic improvements, which are mostly due to H. Iwaniec and E. Kowalski. In particular, for any nontrivial multiplicative character $\chi$ modulo a prime $q$ and any integer $r\ge 2$, we show that $$\sum _{M<n\le M+N}\chi (n)=O(N^{1-1\mathrm{/}r}{q}^{(r+1)\mathrm{/}4r^2}(\log q{)}^{1\mathrm{/}4r}),$$ which sharpens previous results by a factor $(\log q)^{1/4r}$. Our improvement comes from averaging over numbers with no small prime factors rather than over an interval as in previous approaches.