Let \(R(N)\) denote the number of positive integers not exceeding \(N\) which are a sum of three positive integer cubes. It is believed that the integers in question have positive density, but the recorded results are of the form \(R(N) \gg N^{\alpha-\varepsilon}\) for certain \(\alpha < 1\). Using an extension of the “new iterative method” of R. C. Vaughan [Acta Math. 162, 1-71 (1989; Zbl 0665.10033)] the author [Invent.Math. 122, 421-451 (1995; Zbl 0851.11055)], using estimates for fractional moments of exponential sums over smooth numbers, improved Vaughan’s result with \(\alpha={11\over 12}\) to \(\alpha=1-\xi/3\), where \(\xi\) is the positive root of \(\xi^3+16\xi^2+28\xi-8\). The paper under review establishes an improved Mean Value Theorem, which leads to the quadratic irrational value \(\alpha = \bigl(166-\sqrt{2833} \bigr)/123= 0.91686\ldots \). The numerical improvement is very small, occurring in the fifth decimal place.
The conditional value \(\alpha=1\) was obtained by C. Hooley [Acta Math. 157, 49-97 (1986; Zbl 0614.10038); and in Sieve methods, exponential sums, and their applications in number theory (Cardiff, 1995), Cambridge University Press, Lond. Math. Soc. Lect. Note Ser. 237, 175-185 (1997; Zbl 0930.11071)] and by D. R. Heath-Brown [Philos.Trans.R. Soc.Lond., Ser. A 356, No. 1738, 673-699 (1998; Zbl 0899.11051)], but this rests upon a Riemann Hypothesis for certain Hasse-Weil \(L\)-functions that are not known to possess an analytic continuation into the critical strip.
Following an argument of J. Brüdern, the new method slightly improves the estimate for the number \(E(X)\) of positive integers not exceeding \(X\) which are not a sum of four positive cubes, to \(E(X) \ll X^{1-\beta+\varepsilon}\), where \(\beta=\bigl(422-6\sqrt{2833} \bigr)/861 =0.11921\ldots \).
The author’s new mean value theorem refers to \(f(t,P,R) = \smash{\sum_{x \in {\mathcal A}(P,R)}}e\bigl(t x^3\bigr)\), in which \({\mathcal A}(P,R)\) is the set of numbers \(\leq P\) all of whose prime factors are \(\leq R\), and to the moments \(U_s(P,R) = \int_{0<t<1}\bigl|f(t,P,R)\smash{\bigr|^s} dt\) when \(s=5\), but the method implies improvements for some other values of \(s\). The method is based on new iterations between admissible exponents \(\mu_s\) in inequalities of the shape \(U_s(P,R) \ll P^{\mu_s+\varepsilon}\).