Notice Board

Lethbridge Number Theory and Combinatorics Seminar

Date: January 27, 2020

Time: 12:00 to 12:50 p.m.

Lecturer(s): Habiba Kadiri (University of Lethbridge)

Location:  W561, University of Lethbridge

Topic: Explicit results about primes in Chebotarev's density theorem

Description:

Let L/K be a Galois extension of number fields with Galois group G, and let C be a conjugacy class in G. Attached to each unramified prime ideal P in the ring of integers of K is the Artin symbol sigma(P), a conjugacy class in G. In 1922 Chebotarev established what is referred to his density theorem (CDT). It asserts that the number of such primes with sigma(P) = C and norm less than x is asymptotically

|C|/|G| times Li(x) as x tends to infinity, where Li(x) is the usual

|C|logarithmic

integral. As such, CDT is a generalisation of both the prime number theorem and Dirichlet's theorem on primes in arithmetic progressions.

In light of Linnik's result on the least prime in anarithmetic progression, one may ask for a bound for the least prime ideal whose Artin symbol equals C. In 1977 Lagarias and Odlyzko proved explicit versions of CDT and in 1979 Lagarias, Montgomery and Odlyzko gave bounds for the least prime ideal in the CDT. Since 2012 several unconditional explicit results of these theorems have appeared with contributions by Zaman, Zaman and Thorner, Ahn and Kwon, and Winckler. I will present several recent results we have proven with Das, Ng, and Wong.

Other Information:

Web page: http://www.cs.uleth.ca/~nathanng/ntcoseminar/