On the representation of numbers in the form \(a_1X_1^2+a_2X_2^2+a_3X_3^2+a_4W^l\). (Q1811391)

This paper considers the number of representations \(R(N)\) of a large integer \(N\) in the form \[ N= a_1 X^2_1+ a_2X^2_2+ a_3 X^2_3+ a_4 W^l, \] where the coefficients \(a_i\) are given positive integers, and the exponent \(l\) is a fixed integer at least \(2\). Previously, the author [Asian J. Math. 4, 885--904 (2000; Zbl 1030.11052)] had considered the case in which the coefficients \(a_i\) are all equal to \(1\), but the more general case cannot be handled in the same way, since one does not have suitable information about representations as \(a_1 X^2_1+ a_2 X^2_x+ a_3 X^2_3\). Instead the paper uses a form of the circle method described by the reviewer [J. Reine Angew. Math. 481, 149--206 (1996; Zbl 0857.11049)], which allows a very precise form of the ``Kloosterman refinement''. The result is that \[ R(N)= \sum_\infty {\mathfrak S}(N) N^{1/2+ 1/l}+ O(N^{1/2+ 1/l}(\log N)^{-\delta},\tag{\(*\)} \] for any positive constant \(\delta< t- 1/\sqrt{2}\). The singular series \({\mathfrak S}(N)\) is discussed in some detail. It is shown in particular that there are situations in which \({\mathfrak S}(N)\) is positive, and yet is too small for \((*)\) to give an asymptotic formula.