Cubic fields, low-lying zeros and the L-functions Ratios Conjecture

Abstract: In this talk I will discuss recent work on the low-lying zeros in the family of $L$-functions attached to non-Galois cubic Dedekind zeta functions. In particular, I will describe the close relation between these low-lying zeros and precise counting results for cubic fields with local conditions. The main application of this investigation is a conditional omega result for cubic field counting functions. I will also discuss the $L$-functions Ratios Conjecture associated to this family of Dedekind zeta functions and the fact that the conjecture in its standard form does not predict all the terms in the family's one-level density of low-lying zeros. This is joint work with Peter Cho, Daniel Fiorilli and Yoonbok Lee.

number theory

Audience: researchers in the topic

London number theory seminar

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