  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 $$
\sum_{M<n\le M+N}\chi(n) = O\left( N^{1-1/r}q^{(r+1)/4r^2}(\log
q)^{1/4r}\right), $$ 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.
