On Vinogradov's mean value theorem

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 equationswith 1 ≥ xi, yi ≥ P for 1 ≥ i ≥ s. 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 writingin which e(α) denotes e2πiα, we observe thatwhere Tk denotes the k-dimensional unit cube, and α = (α1,…,αk).