On Waring's problem for cubes, it is conjectured that every sufficiently large natural number can be represented as a sum of four cubes of natural numbers. Denoting by E(N) the number of the natural numbers up to N that cannot be written as a sum of four cubes, we may express the conjecture as E(N)≪1.

In 1973, Montgomery [12] introduced, in order to study the vertical distribution of the zeros of the Riemann zeta function, the pair correlation function

where w(u) = 4/(4 + u2) and γjj = 1, 2, run over the imaginary part of the nontrivial zeros of ζ(s). It is easy to see that, for T → ∞,

uniformly in X, and Montgomery [12], see also Goldston-Montgomery [7], proved that under the Riemann Hypothesis (RH)

uniformly for X ≤ T ≤ XA, for any fixed A > 1. He also conjectured, under RH, that (1) holds uniformly for Xε ≤ T ≤ X, for every fixed ε > 0. We denote by MC the above conjecture.

Let s, k and n be positive integers and define rs,k(n) to be the number of solutions of the diophantine equation

in positive integers xi. In 1922, using their circle method, Hardy and Littlewood [2] established the asymptotic formula

whenever s≥(k−2)2k−1 + 5. Here , the singular series, relates the local solubility of (1.1). For each k we define to be the smallest value of s0 such that for all s ≥ s0 we have (1.2), the asymptotic formula in Waring's problem. The main result of this memoir is the following theorem which improves upon bounds of previous authors when k≤9.

Let k ≤ 2 be an integer, and set

Ek (X) = |{n ≤ X, n ≠ mk, n not a sum of a prime and a k-th power}|.

We prove that there exists δ = δ(k) > 0 such that Ek (X)Ek(X)≪kX1−δ.

§1. Introduction. The literature on solving a system of linear equations in primes is quite limited, although the multi-dimensional Hardy-Littlewood method certainly provides an approach to this problem. The Goldbach- Vinogradov theorem and van der Corput's proof of the existence of infinitely many three term arithmetic progressions in primes are two particular results in the special case of only one equation. Recently Liu and Tsang [4] studied this case in full generality and obtained a result with excellent uniformity in the coefficients. Almost no other general result has appeared so far, due probably to the fact that such a theorem is clumsy to state.

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, 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 writing

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

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