# Mathematical Sciences Research Institute

Home » Analytic Number Theory Seminar: New bounds for the Chebotarev density theorem

# Seminar

Analytic Number Theory Seminar: New bounds for the Chebotarev density theorem February 28, 2017 (02:00 PM PST - 03:00 PM PST)
Parent Program: Analytic Number Theory MSRI: Simons Auditorium
Speaker(s) Jesse Thorner (Stanford University)
Description No Description
Video
No Video Uploaded
Abstract/Media

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.

No Notes/Supplements Uploaded No Video Files Uploaded