On octonary cubic forms: II

hooley, christopher;

Let f(x) = f(x1, . . . , x8) be an octonary cubic form with rational integral coefficients and nonzero discriminant. Then, on the assumption of a certain type of Riemann hypothesis, we showed
in the first paper with the above title that the indeterminate equation f(l)=0 has a non-zero solution provided that f(x) conform to the (necessary) condition that it have a non-trivial zero in every p-adic field Qp. Owing to the nature of its proof, this result demonstrated only the existence of such zeros and did not provide a quantitative estimate for the number of these lying in expanding regions of a suitable type. In this paper, we supply such an estimate by proving,
on the same hypothesis as before, an asymptotic formula for a weighted number of solutions l of f(l) = 0 which lie in a region of the type l/X − a < 1 for suitable points a. Certain applications
of this theorem are specified.