Incomplete higher-order Gauss sums

We consider the classical incomplete higher-order Gauss sums S-m(B)= Sigma(j=0)(B) exp(2piij(m)/N), 0 less than or equal to B < N - 1, where N is large. In 1976, Lehmer analyzed the beautiful spirals appearing in the directed graph S-m(0) --> S-m(1) --> S-m(2) --> (...) in the complex plane for m = 2. In the process, he obtained sharp uniform estimates for the sums S-2(B). Sullivan and Zannier expanded on Lehmer's work in 1992 by giving an asymptotic formula for S-2(B) valid for B < N(1/2 - epsilon). For general m > 1, the directed graphs still exhibit spirals, but with more complicated behavior than in the quadratic case. We analyze this behavior and in the process obtain sharp asymptotic estimates for certain incomplete Gauss sums S-m (B). (C) 2003 Elsevier Science (USA). All rights reserved.