|Location:||MSRI: Simons Auditorium|
The Chebotarev density theorem describes the distribution of prime ideals of a number field K with a given ``splitting condition'' in a Galois extension. This result subsumes many classical results in analytic number theory, including the distribution of primes, quadratic residues, primes in arithmetic progressions, and primes that split completely in a Galois extension. Lagarias and Odlyzko made this distribution effective, but the effective dependence on the Galois extension is prohibitive in many applications.
I will discuss new upper and lower bounds for the Chebotarev prime counting function that result from an effective and explicit log-free zero density estimate for Hecke L-functions. One application is a new explicit bound for the least prime ideal in the Chebotarev density theorem which is commensurate with Linnik's bound on the least prime in an arithmetic progression. Another application is an altogether new upper bound for the Chebotarev prime counting function which is commensurate with the Brun-Titchmarsh theorem. We will describe applications to the study of elliptic curves, modular forms, and binary quadratic forms. This is joint work with Asif Zaman.