Distribution problems associated to zeta functions and invariant theory.

by Andrew Yang

We investigate two questions which are natural generalizations of the work of H. Davenport on sums of class numbers of binary cubic forms, and Davenport and H. Heilbronn on asymptotics for the number of cubic number fields of absolute discriminant less than X, as X → infinity. The latter of these questions involves the distribution of the low-lying zeros of Dedekind zeta functions associated to cubic number fields, in the sense of the Katz-Sarnak philosophy. The core of this question is reduced to understanding how rational primes split in number fields K as the field varies. In addition, we require a recent result of K. Belabas, M. Bhargava, and C. Pomerance on power-saving error terms for the number of cubic number fields of absolute discriminant up to X, as X → infinity. Furthermore, we use the results of Bhargava on counting S 4 quartic number fields, which is a natural extension of Davenport and Heilbronn's work, to investigate the low-lying zeros of Dedekind zeta functions attached to S4 quartic number fields. The other question we consider is how to count equivalence classes of integral binary forms of degree higher than three. We briefly examine a reduction theory for such forms developed by G. Julia, which involves the notion of a (nonrational) invariant he calls '&thgr;', and apply this to understanding sums of various types of class numbers. Explicit calculations are possible in the case of cubic and quartic binary forms, and we make a comparision of &thgr; to the more well-known rational invariants of those forms in these cases.

99 pages, NOOKstudy eTextbook