Sums of four cubes of primes. (Q1869786)

Let \(f(N)\) denote the number of positive integers not exceeding~\(N\) which are a sum of four cubes of primes. The author showed fairly recently [Chin. Ann. Math., Ser.~B~22, No.~2, 233--242 (2001; Zbl 0984.11048)] that \(F(N) \geq \beta N\), where \(\beta>0\) is an absolute constant and \(N\)~is to be taken sufficiently large. In the paper under review he gives an admissible value for~\(\beta\), namely \(1\over320\). This implies that at least 1.5\ representable in the desired form. As is usual in work of this nature, a key step is an estimate for a sum \(\sum r^2(l)\), where \(r(l)\) denotes the number of representations of~\(l\) in the required form, subject to certain technically convenient restrictions. This is obtained via the circle method, where the author draws from a paper by \textit{J.~Brüdern} [Ann. Sci. Éc. Norm. Supér. (4) 28, No.~4, 461--476 (1995; Zbl 0839.11045)] in which one of the four cubes involved may be of an almost-prime. Following Brüdern's overall strategy, an appeal is also made to a sieve, for which the author now uses a lemma from a paper of \textit{K.~Kawada} and \textit{T.~D.~Wooley} [Proc. Lond. Math. Soc. (3) 83, No.~1, 1--50 (2001; Zbl 1016.11046)] concerning the representation of numbers as sums of fourth or fifth powers of primes.