Note on Heath-Brown’s estimate for Heilbronn’s exponential sum

Abstract

We show that S h ( a ) = ∑ n = 1 p e ( a n h p p 2 ) ≪ ( h , p − 1 ) p 11 / 12 S_h(a)=\sum ^p_{n=1}e(\frac {an^{hp}}{p^2})\ll (h,p-1)p^{11/12} , which generalizes Heath-Brown’s estimate for Heilbronn’s exponential sum S 1 ( a ) S_1(a) . We also give a simple proof of a crucial lemma in Heath-Brown’s work.