Major arcs in the four cubes problem

Abstract

There is an old conjecture that every integer can be decomposed into four (positive or negative) perfect cubes. More specifically one would like to know the asymptotic number of solutions of when a large bound N is placed on the parts TW, -. Using the circle method it is shown that the number of such representations of n when N -> oo is asymptotically equal to C(«). N for a certain positive constant C(n), provided that the contribution of the minor arcs can be neglected.The dependence of C(«) on n is exhibited explicitly by expressing C(«) as an infinite product. The formula is of heuristic value only since the minor arcs cannot be handled at present.