A N A L Y T I C      N U M B E R    T H E O R Y ,     2 0 2 2

Lectures: Tuesday, Thursday, 2:00pm-3:15pm, in Warren Weaver Hall 512.

Lecturer: Paul Bourgade, office hours Friday 9.30-10.30am, you also can email me (bourgade@cims.nyu.edu) to set up an appointment or just drop by (WWH 603).

Course assistant: Sujay Kazi (ssk9840@nyu.edu). Sections: Friday 12.30-1.45pm, 194M 304.

Course description: An introduction to analytic methods in number theory. Some goals are the proof of Prime Number Theorem using the Riemann zeta function, Dirichletâ€™s theorem on prime numbers in an arithmetic progression, sieve methods. Mathematical technique is developed as needed. This includes basics of complex function theory and integration, Fourier analysis, and finite abelian groups (for Dirichletâ€™s theorem).

Prerequisites: Students must have Analysis I or specific permission of the instructor. Each of these is helpful but not required: Algebra I, Theory of Numbers, Complex Variables.

Textbooks: Our reference text will be The Distribution of Prime Numbers, by Dimitris Koukoulopoulos. An online version can be found here.

Homework: Every Thursday for the next Thursday.

Grading: problem sets (50%) and final (50%).

A tentative schedule for this course is:

• Jan. 25. Introduction, Fundamental theorem of Arithmetic, Prime number theorem and its heuristics, Dirichlet's theorem for primes in arithmetic progressions.
• Jan. 27. Real analysis tools.
• Feb. 1. Complex analysis tools.
• Feb. 3. Combinatorial ways to count primes I.
• Feb. 8. Combinatorial ways to count primes II.
• Feb. 10. The Dirichlet convolution.
• Feb. 15. Dirichlet Series.
• Feb. 17. The Riemann zeta function: definitions, analytic continuation.
• Feb. 22. The Riemann zeta function: functional equation.
• Feb. 24. The Riemann zeta function: heuristics for analytic ways to count primes.
• Mar. 1. A priori estimates on the Riemann zeta function.
• Mar. 3. Proof of the prime number theorem I.
• Mar. 8. Proof of the prime number theorem II.
• Mar. 10. Dirichlet characters.
• Mar. 22. Fourier analysis on finite abelian groups I.
• Mar. 24. Fourier analysis on finite abelian groups II.
• Mar. 29. Dirichlet L-functions I.
• Mar. 31. Dirichlet L-functions II.
• Apr. 5. The prime number theorem in arithmetic progressions I.
• Apr. 7. The prime number theorem in arithmetic progressions II.
• Apr. 12. The Erdos-Kac Theorem.
• Apr. 14. The Selberg Delange method I.
• Apr. 19. The Selberg Delange method II.
• Apr. 21. Twin primes.
• Apr. 26. Sieve theory, axioms.
• Apr. 28. Sieve theory, the fundamental lemma I
• May. 3. Sieve theory, the fundamental lemma II.
• May. 5. Sieve theory, applications.

Problem sets.
• Feb. 3. Exercises 1.1, 1.7, 1.11 here.
• Feb. 10. Exercises 2.3, 2.5, 2.11 here.
• Feb. 17. Exercises 3.8, 3.9, 4.8, 4.9 here.
• Feb. 24. Exercises 1.10 (a) (b), 5.1, 5.3, 6.6 here.
• Mar. 3. Exercises 5.2, 6.2, 7.7 here.
• Mar. 10. Exercises 7.5, 8.5 (a) (b), 8.8 here.
• Mar. 24. Exercises 9.1, 9.3, 10.1 here.
• Mar. 31. Exercises 9.6, 10.3 here.
• Apr. 7. Exercises 10.4, 11.2 here.
• Apr. 14. Exercises 15.1, 15.4, 15.5 here.
• Apr. 21. Exercises 13.1, 13.4, here, and write a self-contained proof of Theorem 13.2 in the special case J=1.
• Apr. 28. Exercises 17.1, 17.3, from here. Also write the shortest, cleanest possible proof of 15.4 (a).