An asymptotic approximation for incomplete Gauss sums

The classical incomplete Gauss sum ∑ j = 0 m - 1 exp ( 2 π i j p / N ) ( 1 ⩽ m < N ; p > 1 ) is studied for large N. An expansion is derived for this sum when p is an integer that holds uniformly for 1 ⩽ m < M 0 , M 0 = ( N / p ) 1 / ( p - 1 ) corresponding to the dominant, primary spiral when the terms are considered as unit vectors in the complex plane. This result is specialised to the quadratic incomplete Gauss sum ( p = 2 ) for which the spirals consist of regular traces dependent on the residue of N (mod 4). This expansion complements earlier work by Lehmer (Mathematika 23 (1976) 125) and extends the more recent results of Evans et al. (J. Math. Anal. Appl. 281 (2003) 454). The above expansion in the primary spiral is also discussed in the case p > 1 . Numerical results are given to demonstrate the accuracy of the various approximations.