Incomplete Gauss sums

Let N be a positive integer. We are concerned with the sumG N ( m ) = ∑ j = 0 m − 1e { 2 π i j 2 / N }.Thus G N ( N ) is the ordinary Gauss sum. Previous methods of estimating such exponential sums have not brought to light the peculiar behaviour of G N ( m ) for m < N /2, namely that, for almost all values of m , G N ( m ) is in the vicinity of the pointN ( 1 + i ) / 4 . A sharp estimate is given for max | G N ( m )|, depending on the residue of N modulo 4. The results were suggested by graphs of G N ( m ) made for N near 1000. The analysis employs the Fresnel integrals and the Cornu spiral whose curvature is proportional to its arc length.