A N A L Y T I C     N U M B E R     T H E O R Y ,    S p r i n g    2 0 1 3

Lectures: Tuesday and Thursday, 11.30am-1pm, in Science Center 216.

Lecturer: Paul Bourgade, office hours Wednesday 4-6pm, you also can email me (bourgade@math.harvard.edu) to set up an appointment or just drop by (Science Center 341).

Course description: topics will include the analytic and elementary proofs of the prime number theorem, Dirichlet L-functions, primes in arithmetic progressions, sieve methods and Stepanov's method for the Weil conjectures in the case of hyperelliptic curves.

Prerequisites: complex analysis.

Textbooks: our reference texts will be Multiplicative Number Theory I Classical Theory, by Montgomery-Vaughan, and Analytic Number Theory by Iwaniec-Kowalski.

Homework: every Tuesday for the next Tuesday.

Grading: problem sets (60%), and a final project (40%).

A tentative schedule for this course is:

• Jan. 29. Arithmetic functions, Dirichlet series.
• Jan. 31. The Riemann ζ  function: functional equation, analytic extension.
• Feb. 5. The Riemann ζ  function: a zero-free domain.
• Feb. 7. Perron's formula, the analytic proof of the Prime Number Theorem.
• Feb. 12. Selberg's proof of the Prime Number Theorem.
• Feb. 14. The explicit formula.
• Feb. 19. The Weierstrass-Hadamard product, general distribution of the ζ  zeros.
• Feb. 21. Primitive charcters and Gauss sums.
• Feb. 26. Analytic properties of Dirichlet L-functions.
• Feb. 28. Primes in arithmetic progressions.
• March 5. Sieve problems.
• March 7. Selberg's Λ² sieve.
• March 12. The large sieve I.
• March 14. The large sieve II.
• March 26. Exponential sums: Weyl.
• March 28. Exponential sums: Van der Corput.
• April 2. Exponential sums: Vinogradov.
• April 4. Sums over primes.
• April 9. The Bombieri-Vinogradov theorem.
• April 11. Dirichlet polynomials.
• April 16. Zero density estimates.
• April 18. Sums over finite fields: introduction.
• April 23. Sums over finite fields: Kloosterman sums.
• April 25. Sums over finite fields: Stepanov's method for hyperelliptic curves.
• April 30. Sums over finite fields: applications.

• Problem set 1.
• Problem set 2.
• Problem set 3.
• Problem set 4.