Sep 1, 2022 — Dec 31, 2022

Instructor
Habiba Kadiri : habiba dot kadiri at uleth dot ca

Prerequisites
Elementary Number Theory, Real and Complex Analysis

Abstract
This is a first course in analytic number theory. In this course we will focus on the theory of the Riemann zeta function and of prime numbers. The goal of this course will include proving explicit bounds for the number of primes which are less than a given number. Building analytical tools to prove the prime number theorem will be at the core of this course. We will explore and compare explicit formulas, whether they are using smooth weights or a truncated Perron formula, to relate averages over primes and to sums over the zeros of zeta. Another originality of this course will be to explore each topic explicitly (essentially by computing all the hidden terms implied in the asymptotic estimates). With this respect, students will get an introduction to relevant programming languages and computational software.
This will be closely connected to Analytic Number Theory 2 by Greg Martin (UBC), as the sequences of topics are coordinated between us; the intention is for students at all PIMS institutions to be able to take the second analytic number theory course as a continuation of the first one with maximum benefit. In addition, these two courses will provide excellent training for students who plan to attend the “Inclusive Paths in Explicit Number Theory” CRG summer school in 2023. All these events are part of the PIMS CRG “L-functions in Analytic Number Theory” (2022-2025).

Syllabus

Course page (access restricted to course participants)

Jan 1, 2023 — Apr 30, 2023

Instructor
Greg Martin : gerg at math dot ubc dot ca

Prerequisites
A course in analytic number theory that includes Dirichlet series and a complex-analytic proof of the prime number theorem (preferably Analytic Number Theory I taught by Kadiri in Fall 2022), or other exposure to those topics

Abstract
This course is a second graduate course in number theory, designed to follow Analytic Number Theory I taught by Prof. Habiba Kadiri in Fall 2022 at the University of Lethbridge. We will learn about Dirichlet characters and sums involving them, Dirichlet L-functions and their zeros, and the prime number theorem in arithmetic progressions. With the explicit formula for the error term in this theorem, we will continue into limiting distributions of error terms and comparative prime number theory (“prime number races”). This course also precedes the summer school “Inclusive Paths in Explicit Number Theory” in Summer 2023 and is designed to give students the ideal preparation for that summer school program. All three of these events are part of the current PIMS CRG “L-functions in Analytic Number Theory” (2022-2025).

Official course web page

Sept. 6, 2023 — Dec. 5, 2023

Instructor
Alia Hamieh : alia dot hamieh at unbc dot ca

Prerequisites
A graduate course in analytic number theory that includes Dirichlet series and a complex-analytic proof of the prime number theorem (preferably Analytic Number Theory I taught by Kadiri in Fall 2022).

Abstract
This course  will cover advanced topics on moments of L-functions. It is intended to follow Analytic Number Theory I taught by Prof. Habiba Kadiri (University of Lethbridge) in Fall 2022 and Analytic Number Theory II taught by Prof. Greg Martin (UBC) in Winter 2023. All three of these events are part of the current PIMS CRG “L-functions in Analytic Number Theory” (2022-2025).

In this course, we will explore advanced topics in moments of L-functions including approximate functional equations, zero density estimates, mean value estimates for Dirichlet polynomials, large sieve inequalities, Poisson and Voronoi summation formulae, shifted convolution sums, holomorphic modular forms and associated L-functions, trace formulae and the spectral theory of automorphic forms. 

Course Webpage