Density of discriminants of cubic extensions.

In this paper, we shall generalize the results of Davenport and Heilbronn in [11] on densities of discriminants of cubic fields and the mean 3-class-number of quadratic fields to the case when the base field is an arbitrary global field of characteristic not equal to 2 or 3. Our methods are based on the adelization of Shintani's work on the zeta functions associated to the space of binary cubic forms. The theory of the adelic zeta function is presented in [7], [8], and [26]. Standard Tauberian theorems may be used to extract the mean value of the class-numbers of S-integral binary cubic forms from the relevant Dirichlet series studied in Section 6 of [8]. Davenport, having obtained the mean value of these class-numbers over Q by means of reduction theory, then, together with Heilbronn, applied a delicate sieve to obtain the density of discriminants results. Our formulation, however, allows us to proceed directly to the question of density of discriminants with no further mention of classes of binary cubic forms. It is worth mentioning at the outset, though, how binary cubic forms are related to cubic extensions. Vectors x = (x1? x2, x3, x4) are identified with binary cubic forms s follows Fx(u, v) =

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