Lecture: Monday-Wednesday 8:30am‑9:50am pacific time on Zoom (request zoom link from instructor by email)
Instructor: Alia Hamieh
Office Hours: by appointment
This course 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 courses are part of the current PIMS CRG “L-functions in Analytic Number Theory”.
In this course, we will explore advanced topics in moments of L-functions. In the first part of the course, we will focus on moment results for the Riemann zeta function. We will cover all the machinery required to establish an asymptotic formula for the fourth moment of the Riemann zeta function on the critical line following the original work of Ingham while also discussing Heath-Brown’s improvements on Ingham’s result. In the second part of the course, we will introduce automorphic L-functions which generalize the class of degree 1 L-functions to which the Riemann zeta function belongs. We study their basic properties, explicit formulae, approximate functional equations and zero free regions. We will then focus on classical automorphic L-functions associated to the automorphic forms on GL(2) (that are either holomorphic or eigenfunctions of the Laplace operators). We will learn about the necessary tools needed to study moments of families of such L-functions such as Voronoi summation formula, Petersson and Kustensov trace formulae, and estimates for shifted convolution sums. We will establish asymptotic formulae for second moments of families of degree 2 automorphic L-functions, and discuss results pertaining to higher moments as time permits. We will also discuss applications of moment estimates to obtaining subconvexity bounds and establishing non-vanishing results for the underlying L-functions.
Lecture Notes and Videos
- Week 1: Wednesday Sept 6 lecture video and notes
- Week 2: Monday Sept 11 lecture video and notes
Wednesday Sept 13 lecture video and notes
- Week 3: Monday Sept 18 lecture video and notes
Wednesday Sept 20 lecture video and notes
- Week 4: Monday Sept 25 lecture video and notes
Wednesday Sept 27 lecture video and notes
- Week 5: Monday Oct 2 lecture video and notes
Wednesday Oct 4 lecture video and notes (Guest lecture by Prof. Nathan Ng — Starts at 8am pacific time)
Title: Approximate Functional Equations
Abstract: I will give an overview of the history of the approximate functional equation and the Riemann-Siegel formula.
I will discuss smooth approximate functional equations and show some applications.
- Week 6: Wednesday Oct 11 lecture video and notes
- Week 7: Monday Oct 16 lecture video and notes
Wednesday Oct 18 lecture video and notes (Guest lecture by Dr. Sieg Baluyot)
Title: The CFKRS recipe for moments of L‑functions.
Abstract: We will discuss the well-known heuristic due to Conrey, Farmer, Keating, Rubinstein, and Snaith for predicting
asymptotic formulas for moments of L‑functions. We will carry out their heuristic for different examples of families of L‑functions,
and compare their conjectures with predictions from random matrix theory. The main reference
- Week 8: Monday Oct 23 lecture video and notes
Wednesday Oct 25 lecture video and notes
- Week 9: Monday Oct 30 lecture video and notes
Wednesday Nov 1 lecture video and notes
- Week 10: Monday Nov 6 lecture video and notes and slides
Thursday Nov 9 lecture video and notes (Guest lecture by Dr. Caroline Turnage-Butterbaugh)
Title: The Asymptotic Large Sieve and Moments of Dirichlet L‑functions
Abstract: I will begin with a brief overview of the moment problem in the context of the Riemann zeta-function, and then shift
to consider the problem for the family of all Dirichlet L‑functions of even primitive characters of bounded conductor. I will
outline how to harness the asymptotic large sieve to prove an asymptotic formula for the general 2kth moment of
an approximation to this family. The result, which assumes the generalized Lindelöf hypothesis for large values of k, agrees
with the prediction of Conrey, Farmer, Keating, Rubenstein, and Snaith. Moreover, it provides the first rigorous evidence
beyond the so-called “diagonal terms” in their conjectured asymptotic formula for this family of L-functions.
- Week 11: Monday Nov 13 (class cancelled)
Wednesday Nov 15 lecture video and notes (Guest Lecture by Dr. Alexandra Florea)
Title: Negative moments of L‑functions
Abstract: We will discuss conjectures concerning negative moments of the Riemann zeta- functions, based on number theoretic
heuristic arguments (due to Gonek) and based on random matrix theory (due to Berry and Keating). We will also explain why the
problem of rigorously computing negative moments is more delicate than that of computing positive moments, and will then
describe some recent progress towards this question.
- Week 12: Monday Nov 20 lecture video and notes
Wednesday Nov 22 lecture video and notes