# MATH 740: Moments of L‑functions

Lec­ture: Mon­day-Wednes­day 8:30am‑9:50am pacif­ic time on Zoom (request zoom link from instruc­tor by email)
Instruc­tor: Alia Hamieh
E‑mail: alia.hamieh@unbc.ca
Office Hours: by appoint­ment
Course Out­line

Course Descrip­tion:

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 NgStarts 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
is https://doi.org/10.1112/S0024611504015175
• 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
• Week 13: Mon­day Nov 27 lec­ture video and notes
Wednes­day Nov 29 lec­ture video and notes (Guest Lec­ture by Dr. Kevin KwanStarts at 8am pacif­ic time)
Title: Anoth­er look at the clas­si­cal moments
Abstract: In this lec­ture, we will explore how the eigen­func­tions of the Lapla­cian and their peri­ods enter the study of cer­tain
moments of the zeta-func­tion and the Dirich­let L‑functions.
• Week 14: Mon­day Dec 4 lec­ture video and notes