New Bounds for Gauss Sums Derived From k-th Powers, and for Heilbronn’s Exponential Sum

where p is prime, e(x) = exp(2πix), and ep(x) = e(x/p). In each case we shall assume that p | / a unless the contrary is explicitly stated. Gauss sums arise in investigations into Waring’s problem, and other additive problems involving k-th powers. Although they are amongst the simplest complete exponential sums, the question as to their true order of magnitude is far from being resolved. We remark at the outset that if (k, p− 1) = k0, then Gp(a, k) = Gp(a, k0). Thus it suffices to suppose, as indeed we shall, that k|p− 1. When p | / a the trivial bound for G(a) states that |G(a)| ≤ p. The next simplest estimate takes the form |G(a)| ≤ (k − 1)√p. (1) This may be obtained by writing G(a) in terms of the character Gauss sum as