New bounds for Gauss sums derived from KTH powers, and for Heilbronn's exponential sum

We show that p/Σ/n=1 exp (2π ian^k/p) << min (k^5/8p^5,8, k^3/8p^3/4and p/Σ/n=1 exp (2π ian^p/p²) << p^7/8 when p Xa. The proof uses a modification of Stepanov's method.