An asymptotic approximation for incomplete Gauss sums. II

An expansion for the incomplete Gauss sum S m ( x ; p ) = ∑ j = 0 m - 1 exp ( π i xj 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].