An asymptotic approximation for incomplete Gauss sums. II

An expansion for the incomplete Gauss sum S-m (x; p) = (m-1)Sigma(j=0) exp(pi ixj(p)), p > 1 is obtained for x -> 0(+) for values of m corresponding to the principal spiral 1 <= m < M-0, M-0 = (2/px)(1/(p-1)) (when the terms of the sum are considered as unit vectors in the complex plane). This expansion results from resumming the terms in the expansion obtained in Paris [An asymptotic approximation for incomplete Gauss sums, J. Comput. Appl. Math. 180 (2005) 461-477]. The new expansion is specialised to the quadratic incomplete Gauss sum with p = 2 and x = 2/N, where N is a large positive integer, and compared with that obtained by Evans et al. [incomplete higher-order Gauss sums, J. Math. Anal. Appl. 281 (2003) 454-476]. (C) 2006 Elsevier B.V. All rights reserved.