On the representations of numbers by binary cubic forms

Abstract

There are not a few situations in the theory of numbers where it is desirable to have as sharp an estimate as possible for the number r(n) of representations of a positive integer n by an irreducible binary cubic formA variety of approaches are available for this problem but, as they stand, they are all defective in that they introduce unwanted factors into the estimate. For instance, an estimate involving the discriminant of f(x, y) is obtained if we adopt the Lagrange procedure [5] of using congruences of the type f(σ, 1)≡0, mod n, to reduce the problem to one where n=1. Alternatively, following Oppenheim (vid. [2]), Greaves [3], and others, we may appeal to the theory of factorization of ideals, which leads to unwanted logarithmic factors owing to the involvement of algebraic units. Having had need, however, in some recent work on quartic forms [4] for an estimate without such extraneous imperfections, we intend in the present note to prove thatuniformly with respect to the coefficients of f(x, y), where ds(n) denotes the number of ways of expressing n as a product of s factors.