On the sum of four cubes and a product of \(k\) factors (Q1922169)

Let \(R_k(N)\) denote the number of representations of \(N\) as a sum of four positive cubes and a product of \(k\) factors. The author's result, obtained using the circle method, implies that provided \(k\geq 3\) the asymptotic formula \[ R_k(n)= N^{4/3}(\log N)^{k-1}(P(1/\log N)+E(N)); \qquad E(N)\ll 1/(\log n)^{3+\varepsilon} \] holds, where \(P\) is a certain explicitly described polynomial. In the case \(k=3\) such a result (with a weaker expression for \(E(N)\)) was provided by \textit{C. Hooley} [Recent progress in analytic number theory, Vol. 1, 127-191, Academic Press (1981; Zbl 0463.10037)], using not the circle method but ideas also applicable [see \textit{C. Hooley}, J. Reine Angew. Math. 328, 161-207 (1981; Zbl 0463.10036)] to the case \(k=2\). The current treatment depends on a treatment of the minor arcs in the circle method introduced by \textit{R. C. Vaughan} [J. Reine Angew. Math. 365, 122-170 (1986; Zbl 0574.10046)] in connection with the eight cubes problem. The author also obtains an asymptotic formula in the problem considered by \textit{H. Davenport} [Acta Math. 71, 123-143 (1939; Zbl 0021.10601)] of representing a number as a sum of four cubes and a prime.