Burgess’s Bounds for Character Sums

Abstract

Let $S(N;H) =\sum _{N<n\leq N+H}\chi (n)$ be a character sum to modulus q. Then the standard Burgess bound takes the form $S(N;H) \ll _{\varepsilon,r}B_{r}$, where $B_{r} = {H}^{1-1/r}{q}^{(r+1)/4{r}^{2}+\varepsilon }$. We show that

$$\displaystyle{\sum _{j=1}^{J}\max _{ h\leq H}\vert S(N_{j};h){\vert }^{3r} \ll _{\varepsilon,r}B_{r}^{3r}}$$

for any positive integers N _j ≤ q spaced at least H apart, so that even reducing to a single term of the sum recovers the Burgess estimate.