Andrew Yang (Dartmouth)
Low-lying zeros of Dedekind zeta functions attached to cubic number
The Katz-Sarnak philosophy asserts that to any "naturally defined
family" of L-functions, there should be an associated symmetry group
which determines the distribution of the low-lying zeros of those
L-functions. We consider the family of Dedekind zeta functions of cubic
number fields, and we predict that the associated symmetry group is
symplectic. To analyze the low-lying zeros of this family, we start by
using (as is standard in these types of problems) a variant of the
explicit formula used by Riemann to study the Riemann zeta function.
This reduces the problem to understanding the distribution of how
rational primes split in cubic fields of absolute discriminant X, as X
tends to infinity. This can be analyzed by using the work of H.
Davenport and H. Heilbronn on the asymptotics of the number of cubic
fields as the absolute discriminant tends to infinity. The final
ingredient is a recent result of K. Belabas, M. Bhargava, and C.
Pomerance on power-saving error terms in the count of cubic fields
considered by Davenport and Heilbronn. If time permits we will discuss
the adjustments that can be made to this argument to prove a similar
result for S4 quartic number fields, using Bhargava's
generalization of the work of Davenport and Heilbronn.