On Waring's problem for cubes

Abstract

A classical conjecture in the additive theory of numbers is that all sufficiently large natural numbers may be written as the sum of four positive cubes of integers. This is known as the Four Cubes Problem, and since the pioneering work of Hardy and Littlewood one expects a much more precise quantitative form of the conjecture to hold. Let v(n) be the number of representations of n in the proposed manner. Then the expected formula takes the shapewhere (n) is the singular series associated with four cubes as familiar in the Hardy–Littlewood theory.