Research Article

On Vinogradov's mean value theorem

Trevor D. Wooleya1

a1 Department of Mathematics, University of Michigan Ann Arbor, MI 48109-1003 U.S.A..

The object of this paper is to obtain improvements in Vinogradov's mean value theorem widely applicable in additive number theory. Let Js,k(P) denote the number of solutions of the simultaneous diophantine equations


with 1 ≥ xi, yiP for 1 ≥ is. In the mid-thirties Vinogradov developed a new method (now known as Vinogradov's mean value theorem) which enabled him to obtain fairly strong bounds for Js,k(P). On writing


in which e(α) denotes e2πiα, we observe that


where Tk denotes the k-dimensional unit cube, and α = (α1,…,αk).

(Received September 09 1990)

Key Words:

  • 11P05: NUMBER THEORY; Additive number theory, partitions; Waring's problem.