On Waring's problem for three squares and an \(\ell\)th power (Q5947042)

Let \(\nu(n)\) denote the number of representations of \(n\) as a sum of three squares and an \(l\)th power, where \(l >2\). The author establishes an asymptotic formula for \(\nu(n)\) valid as \(n \to \infty\), subject to a certain qualification when \(l\) is even. The formula reads \(\nu(n) \sim C n^{1/2+1/l}{\mathfrak S}(n)\), where \(C\) is expressed in terms of the \(\Gamma\)-function and \(\mathfrak S\) denotes a singular series of the type familiar in the circle method. The qualification is that if \(2\mid l\) then there must exist a fixed \(\delta>0\) such that \(2^\beta < n^\delta\), in which \(n=2^\beta n_1\) with \(n_1\) odd, but the author sketches how the treatment can be extended so as to lead to a formula valid as \(n_1 \to \infty\). NEWLINENEWLINENEWLINEThe method of proof does not, however, employ the circle method; the author remarks that the problem does not appear to fall within the current ambit of that method when \(l>2\). He expresses \(\nu(n)\) in terms of \(\sum_W r\bigl(N-W^l\bigr)\), where \(r(n)\) is the number of representations of \(n\) as a sum of three squares, which is in turn expressed in terms of a hybrid of the two classical approaches using Dirichlet series and Gauss sums, respectively. NEWLINENEWLINENEWLINEThe paper under review was announced as ``to appear'' in the author's survey [Proc. Int. Congr. Math., Warszawa 1983, Vol. 1, 85-97 (1984; Zbl 0565.10047)]. The author reports that at that time his methods dealt successfully only with the case \(l \leq 7\). The method now employed also refers to a theorem of \textit{E. Bombieri} and \textit{J. Pila} [Duke Math. J. 59, 337-357 (1989; Zbl 0718.11048)].