Estimates for complete multiple exponential sums

where the sum is taken over a complete set of residues for x modulo q and eq(t) = e2πit/q. The study of these sums is readily motivated by applications in analytic number theory and elsewhere. The first important estimates for sums in one variable appear in the work of Weyl (1916) on uniform distribution. This led to van der Corput’s method with applications to the zeta function, the divisor problem and other problems in multiplicative number theory. Multiple exponential sums first appeared in work on the Epstein zeta function by Titchmarsh (1934). (Graham and Kolesnik (1991) discuss the history and recent results.) On the other hand, and of more immediate relevance to what follows, Hardy and Littlewood (1919) found a new method for tackling problems in additive number theory such as the problems of Waring and Goldbach. The treatment of the major arcs by this method involves complete exponential sums. (See, for example, Vaughan (1981).) As a consequence of his proof of the Weil conjectures, Deligne (1974) showed that, for a prime p, |S(f ; p)| ≤ (d− 1)npn/2,