On the number of solutions of the diophantine equation \(ax_ 1x_ 2+bx_ 3x_ 4=N\) (Q807661)

An asymptotic formula is derived, by a variant of the circle method due to Voronin, for the number of solutions of the equation mentioned in the title of the paper, where N,a,b, and the \(x_ i\) are natural numbers. This formula holds uniformly for \(ab\ll N^{1/4}\), and its error term is \(O(\tau (N)N^{3/4}\log^ 7N)\), where \(\tau\) (N) denotes the number of divisors of N. In the classical case \(a=b=1\), such a result is known to follow if Weil's estimate for Kloosterman sums is used in an argument due to Estermann. It should be noted that Motohashi has quite recently established - in this special case - the improved estimate \(O(N^{0.7+\epsilon})\) by spectral methods.