On Waring's problem: two squares, two cubes and two sixth powers (Q2922423)

The object of this paper is to prove an asymptotic formula for representations of a large positive integer \(n\) as a sum of \(2\) squares and \(4\) cubes, with two of them being indeed also squares. Let \(E(X,f)\) be the number of integers for which the distance of the number \(R_s(n)\) of representations of \(n\) as a sum of \(2\) squares and \(2\) cubes and \(2\) sixth powers to a special product involving the singular series exceeds \(n/f(n)\) where \(f(t)\) is a function defined over positive \(t\) that grows monotonically to infinity and that when \(t\) is large one has \(f(t) = O(t^{\delta})\) for an appropriate \(\delta\) depending on the context. The author proves using the circle method: NEWLINE\[NEWLINE E(X,f) \ll \ln(X)^3 f(X)^2, NEWLINE\]NEWLINE with the concrete choice \(f(n)=\ln(\ln(n))\); this means that for each positive epsilon the corresponding asymptotic formula fails for at most \(O(\ln(X)^{3+\varepsilon})\) of the integers \(n\) with \(1 \leq n \leq X.\)