- 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
*Analytic continuation of Dirichlet L-functions & the Mellin transform*, by Achintya Polavarapu and Henry Twiss*General structure of Mellin transforms*, by Andreas Hatziiliou and Arnab Bose*The Pólya–Vinogradov inequality*, by Shubhrajit Bhattacharya and Reginald Simpson*Euclidean style proof of Dirichlet's theorem*, by Sreerupa Bhattacharjee and Kin Ming Tsang*On Siegel exceptional zeros and Siegel's theorem*, by Zhengheng Bao and Tan Vo
**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- Friday, March 24: video and notes
- Wednesday, March 22: video and notes
- Monday, March 20 (guest lecturer Youness Lamzouri): video and notes
**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- Friday, March 17: video and notes
- Wednesday, March 15: video and notes
- Monday, March 13: video and notes
**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*X*^{π}(*q*;*a*,*b*), including second-order terms that depend on*a*and*b*as well as*q*- Friday, March 10: video and notes
- Wednesday, March 8: video and notes
- Monday, March 6 (guest lecturer Daniel Fiorilli): notes
**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- Friday, March 3: video and notes
- Wednesday, March 1: video and notes
- Monday, February 27 (guest lecturer Nathan Ng): video and notes
**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- Friday, February 17: video and notes
- Wednesday, February 15 (guest lecturer Lucile Devin): video and notes
- Monday, February 13: video and notes
**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- Friday, February 10: video and notes
- Wednesday, February 8: video and notes
- Monday, February 6: video and notes
**Week 4**: Functional equation for Dirichlet*L*-functions, zero-free region for Dirichlet*L*-functions, exceptional zeros- Friday, February 3: video and notes
- Wednesday, February 1: video and notes
- Monday, January 30: video and notes
**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- Friday, January 27: video and notes
- Wednesday, January 25: video and notes
- Monday, January 23: video and notes
**Week 2**: Dirichlet characters, Gauss sums, character sums- Friday, January 20 (guest lecturer Leo Goldmakher): video and notes
- Wednesday, January 18: video and notes
- Monday, January 16: video and notes
**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
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
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( 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.
- 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.
- 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.
- 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*
- 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*
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