Lectures: Mondays and Wednesdays, 10:00–11:20 AM, via Zoom

  • Zoom link has been emailed to students (please don't distribute widely)

Office hours: right after class or by appointment, via Zoom
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Lecture notes:

Course description: We will begin with a quick review of the prime number theorem and the “explicit formula”, then develop the theory of Dirichlet characters, and combine these two sets of tools to prove the prime number theorem in arithmetic progressions. We will then move into comparing two counting functions of primes in arithmetic progressions, going through the history of such comparisons and learning how the normalized difference can be modeled by random variables, thus giving us a way to understand its limiting distribution. Student assessment will consist of some modest combination of presentations and reviews of research articles.

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. 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.

Classes will be held live (synchronously) on Zoom and regular attendance will be important. The current tentative schedule is to meet at 10am Pacific time on Mondays, Wednesdays, and Fridays. Students can join from any physical location.

Assessment for this course will be modest—our goal in these challenging times will be to learn as much as we can without creating stress for ourselves. Students will be asked to complete short summaries of several research articles from the literature and to give some short lectures during the semester.

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

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