Lecture Notes

These lecture notes are the only required reading for the course. Homework questions are included in the notes - please see the assignments page to find out when they were assigned.

  • Introduction to the course (PDF)
  • The prime number theorem (PDF)
  • Dirichlet series and arithmetic functions (PDF)
  • Dirichlet characters and L-functions (PDF)
  • Primes in arithmetic progressions (PDF)
  • The functional equation for the Riemann zeta function (PDF)
  • Functional equations for Dirichlet L-functions (PDF)
  • Error bounds in the prime number theorem (PDF)
  • More on the zeroes of zeta (PDF)
  • von Mangoldt's formula (PDF)
  • Error bounds in the prime number theorem in arithmetic progressions (PDF)
  • Revisiting the sieve of Eratosthenes (PDF)
  • Brun's combinatorial sieve (PDF)
  • The Selberg sieve (PDF)
  • Applying the Selberg sieve (PDF)
  • Introduction to large sieve inequalities (PDF)
  • A multiplicative large sieve inequality (PDF)
  • The Bombieri-Vinogradov theorem (statement) (PDF)
  • The Bombieri-Vinogradov theorem (proof) (PDF)
  • Prime k-tuples (PDF)
  • Small gaps between primes (after Goldston-Pintz-Yildirim) (PDF)
    (see also the article by Soundararajan and the article by Goldston, Motohashi, Pintz, and Yildirim)
  • Small gaps between primes (proofs) (PDF)
    (again, see article by Goldston, et al.)
  • Artin L-functions and the Chebotarev density theorem (PDF)
  • Elliptic curves and their L-functions (PDF)
  • The Sato-Tate distribution (PDF)