Department Mathematik
print
Navigationspfad
Inhaltsbereich

Analytic Number Theory

Course (4 hours weekly + 2 hours Problem sessions) by O. Forster
Winter Semester 2001/02, Department of Mathematics, LMU München

Time and Room: Mon 9-11 HS 132, Wed 11-13 E27

Problem sessions: Mo 14-16, HS 132

What this course is about: One of the classical problems in number theory is the distribution of primes. In analytic number theory one uses the theory of analytic functions to attack this and other problems. In this course, we will (among other things) give a proof of the Prime Number Theorem, asserting that the number of primes less than or equal to a bound X is asymptotically equal to X/log(X) as X tends to infinity. The proof uses the Riemann zeta function. We will also explain the famous Riemann Hypothesis about the zeroes of the zeta function (this is one of the Millennium Problems, worth $ 1 million, if you can solve it). Another theme is Dirichlet's theorem on primes in arithmetic progressions, where so called L-functions are used.

Contents

  1. Notations and Conventions (pdf)
  2. Divisibility. Unique factorization theorem (pdf)
  3. Congruences. Chinese remainder theorem (pdf)
  4. Arithmetical functions. Möbius inversion theorem (pdf)
  5. Zeta function. Euler product (pdf)
  6. Euler-Maclaurin summation formula (pdf)
  7. Dirichlet series (pdf)
  8. Group characters. Dirichlet L-series (pdf)
  9. Primes in arithmetic progressions (pdf)
  10. The Gamma function (pdf)
  11. Functional equation of the zeta function (pdf)
  12. The Chebyshev functions theta and psi (pdf)
  13. Laplace and Mellin transform (pdf)
  14. Proof of the prime number theorem (pdf)

Prerequisites: Elements of the theory of analytic functions, including residue calculus.

Bibliography

  • Apostol: Introduction to analytic number theory. Springer
  • Hardy/Wright: An introduction to the theory of numbers. Oxford University Press
  • Rose: A course in number theory. Oxford University Press
  • Hlawka/Schoißengeier/Taschner: Geometric and analytic number theory. Springer
  • Chandrasekharan: Introduction to analytic number theory. Springer
  • Edwards: Riemann's zeta function. Academic Press
  • Landau: Zahlentheorie, Vol. 2. Reprint Chelsea

Vorlesungen vergangener Semester

Bücher/Books    Eprints    Software
Otto Forster 2001-02-16