Burgess’s Bounds for Character Sums
Burgess’s Bounds for Character Sums
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 ≤ …