Lectures: Mondays, Wednesdays, and Fridays 10:00–10:50 AM (see the “Class days” section below for the specific days we will and will not be meeting)

  • Lectures will be physically held in room ESB 4133 (Earth Sciences Building), and will be simultaneously broadcast on Zoom
  • Zoom link has been emailed to students (please don't distribute widely)
Office hours: right after class, or by appointment
  • I can meet with students either physically in room MATH 212 (Mathematics Building), or on Zoom at the same URL as the lectures
Email address: gerg@math.ubc.ca

Student writeups

Lecture videos and notes

  • Week 11: Number-field and function-field analogues of prime number races; results without fully assuming LI; averages/smoothed versions of prime number counting functions and their error terms
    • Friday, March 31: video and notes
    • Wednesday, March 29: video and notes
    • Monday, March 27 (guest lecturer Florent Jouve): video and notes
  • Week 10: Prime number races with three or more competitors; obstacles when not assuming GRH, barriers
  • Week 9: continuing to look at how the densities δ(q;a,b) depend on a and b; Hooley's conjecture; introduction to how three-variable comparisons are more complicated than two-variable comparisons
  • Week 8: asymptotic formula for the density δ(q;a,b) associated with the prime number race between π(x;q,a) and π(x;q,b), detailed evaluation of the variance of Xπ(q;a,b), including second-order terms that depend on a and b as well as q
  • Week 7: the Mertens sum and the problems of Pólya and Turán, Fourier transforms of explicit-formula-like functions and associated random variables, logarithmic densities in prime number races
  • Week 6: Limiting distributions of exponential sums, limiting logarithmic distributions of sums over zeros from explicit formulas, oscillations of error terms associated to prime counting functions
  • Week 5: Bounds on exceptional zeros, the prime number theorem in arithmetic progressions, the explicit formula for ψ(x;q,a), the error term and range of uniformity in the prime number theorem for APs and how exceptional zeros or GRH would affect them, the Bombieri–Vinogradov theorem
  • Week 4: Functional equation for Dirichlet L-functions, zero-free region for Dirichlet L-functions, exceptional zeros
  • Week 3: Dirichlet L-functions (Euler product, half-plane of convergence, value at s=1), Dirichlet's theorem on primes in arithmetic progressions, Poisson summation and theta functions
  • Week 2: Dirichlet characters, Gauss sums, character sums
  • Week 1: Review of the Riemann zeta function and the proof of the prime number theorem, characters on finite abelian groups, introduction to Dirichlet characters

Course description: This course is a second graduate course in number theory, intended to follow Analytic Number Theory I which is taught by Prof. Habiba Kadiri (University of Lethbridge) in Fall 2022. 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 Collaborative Research Group L-functions in Analytic Number Theory.

After a quick review of the prime number theorem and the “explicit formula”, we will start by learning about Dirichlet characters and sums involving them (character sums over intervals and Gauss sums), which is sufficient preparation to prove Dirichlet's theorem on the infinitude of primes in arithmetic progressions. We will then study Dirichlet L-functions and their zeros, which will allow us to prove the stronger prime number theorem in arithmetic progressions, including the explicit formula for the error term. From this result we can continue into limiting distributions of error terms in analytic number theory (learning how they can be modeled by random variables) and describe the foundations of comparative prime number theory, which includes “prime number races” as well as related problems of Mertens, Polyá, and Turán. We will assume, but quickly review as we go, results about arithmetic functions and global prime counting functions, as well as the complex-analytic proof of the prime number theorem itself. As part of the PIMS CRG on L-functions in Analytic Number Theory, the course will include several guest lectures by experts in the field.

Recommended prerequisites are a solid course (preferably graduate-level) in elementary number theory, and a graduate-level course in analytic number theory, one that included a proof of the prime number theorem and the corresponding explicit formula for ψ(x). An undergraduate course in probability would also be helpful. Reference texts would be standard analytic number theory books by Iwaniec & Kowalski, by Montgomery & Vaughan, and by Titchmarsh. Students who are willing to learn some of this background as they go are welcome.

The evaluation for this course will consist of regular attendance and of 1–2 write-ups (5–10 pages) of specific topics or results related to the subject matter, which will be completed either individually or in teams depending on the enrollment in the course.

Western Deans Agreement: UBC students can register for Analytic Number Theory I, and graduate students outside UBC can receive credit for this course, according to the Western Deans Agreement. You will fill out a WDA Authorization and Course Registration form that will be approved by your department, your school's faculty of graduate studies, the math department offering the course, and finally by the faculty of graduate studies of the university offering the course. While you don't have to do much after filling out the form, be aware that these multiple steps will take several days in total.

  • For non-UBC students registering for Analytic Number Theory II (this course, MATH 613D at UBC), we must receive the completed form by January 6, 2023, so I encourage you to fill out the form and start the chain of authorizations by December 15, 2022.
More information about the application process for courses through the Western Deans Agreement can be found on their website. Three further notes:
  • Non-UBC students who take this course will be given a UBC email address—make sure you forward it to an email account you check regularly. Your grade for this course will not appear on your school's transcript (I believe) but rather you will get a separate UBC transcript for this course.
  • The UBC mathematics department has a process by which undergraduate students can apply to take our graduate courses (inquire with the Graduate Program Coordinator in our office). However, I don't believe that undergraduate students are eligible for credit for courses at other universities through WDA. I am certainly open to students unofficially auditing my lectures, and I believe Prof. Kadiri feels the same about Analytic Number Theory I.
  • The Western Deans Agreement applies to students from certain universities (mostly in western Canada). However, other students can also take this course through UBC's Graduate Exchange Agreement or through non-exchange visitor status.
Please don't hesitate to check with me or ask me questions (at gerg@math.ubc.ca) regarding these registration issues—they can be confusing and unclear.

Class days: We will have lectures every Monday, Wednesday, and Friday morning starting on Monday, January 9 and ending on Wednesday, April 12. We will not have lectures during the week of February 20–24 (UBC's midterm break), nor will we have lectures on Friday, April 7 nor Monday, April 10 (UBC closes for certain Christian holidays). Be warned that British Columbia will set its clocks forward one hour on Sunday, March 12 due to Daylight Savings Time; if you live somewhere that does not observe Daylight Savings Time on that day, be prepared for class to start at a different time for you on Monday, March 13.

References for analytic number theory:

  • H. L. Montgomery and R. C. Vaughan, Multiplicative Number Theory I. Classical Theory
  • H. Iwaniec and E. Kowalski, Analytic Number Theory
  • P. T. Bateman and H. G. Diamond, Analytic Number Theory: An introductory course
  • H. Davenport, Multiplicative Number Theory
  • T. M. Apostol, Introduction to Analytic Number Theory

Possible references for elementary number theory:

  • I. Niven, H. S. Zuckerman, and H. L. Montgomery, An Introduction to the Theory of Numbers
  • G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers

Other resources and policies: We are committed to offering an inclusive, collaborative and safe space for everyone involved in this course. We ask that all participants are committed to supporting equity, diversity and inclusion; we also ask that all participants have read the PIMS Code of Conduct and agree to follow it.

UBC provides resources to support student learning and to maintain healthy lifestyles but recognizes that sometimes crises arise, and so there are additional resources to access including those for survivors of sexual violence. UBC values respect for the person and ideas of all members of the academic community. Harassment and discrimination are not tolerated nor is suppression of academic freedom. UBC provides appropriate accommodation for students with disabilities and for religious, spiritual and cultural observances. UBC values academic honesty and students ae expected to acknowledge the ideas generated by others and to uphold the highest academic standards in all of their actions. Details of the policies and how to access support are available here.