An estimate for Heilbronn's exponential sum

for any integer a coprime to p. It is important to note here that if n ≡ n′ (mod p), then n ≡ n′p (mod p). Thus the summand in S(a) has period p with respect to n, so that S(a) is a ‘complete sum’ to modulus p. It was a favourite problem of Heilbronn, and later of Davenport, to show that S(a) = o(p) as p → ∞. Odoni [3] examined sums related to S(a), and showed that they are O(p) in a suitable average sense. However his argument fails to give a non-trivial upper bound for an individual value of S(a). Indeed he shows how Weil’s approach leads only to the estimate S(a) = O(p), which is worse than the trivial bound! We can now answer Heilbronn’s question with the following theorem.