What is zeta in math

The Riemann zeta function

An introduction to analytic number theory

Lecture by O. Forster in the winter semester 2008/09
at the Mathematical Institute of the LMU Munich

Wed 14-16, HS A027, Theresienstr. 39

Exercises for this every 14 days. Fri 14-16, A027

In 1859 B. Riemann published his groundbreaking work `` On the number of prime numbers under a given size ''. The thesis deals with the zeta function and its connection with the function pi (x), which gives the number of prime numbers less than or equal to x. The zeta function is initially defined as the infinite sum of 1 / n ^ s over all natural numbers n. The series converges for real s greater than 1. Riemann also considers the function for complex s and shows that it is analytically integrated into the whole complex plane can be continued with a single pole of the first order at the point s = 1. In addition, Riemann proves a functional equation for the zeta function and suggests that all non-real zeros of zeta (s) have the real part 1/2. Although this was confirmed numerically for more than a billion zeros, the Riemann Hypothesis (which is also one of the Millennium Problems) is still unproven to this day. In this lecture (on the 150th anniversary of the Riemann Hypothesis) we introduce the zeta function, use it to prove the prime number theorem that pi (x) is asymptotically equal to x / log (x) and discuss some implications and equivalences to the Riemann Hypothesis.

For: Students of mathematics in their main course

Previous knowledge: Elements of function theory (up to the residual theorem).
Basic knowledge of elementary number theory is useful, but not essential.

Sham is valid as a half exercise certificate for the main diploma, pure mathematics

Lecture script:

Chapters 1-2 and 6-10 were prepared by Andreas Wadhwa.

  1. Zeta function. Euler product (pdf)
  2. Dirichlet series and arithmetic functions (pdf)
  3. Euler-Mascheronian constant and Dirichletscher partial substitute (pdf)
  4. Equivalences to the prime number theorem (pdf)
  5. Proof of the prime number theorem (pdf)
  6. The gamma function (pdf)
  7. Functional equation of the zeta function (pdf)
  8. The zeros of the zeta function (pdf)
  9. The Lindelöf Hypothesis (pdf)
  10. Equivalences to the Riemann Hypothesis (pdf)
T. Apostol: Introduction to Analytic Number Theory. Springer 1976
J. Brüdern: Introduction to analytic number theory. Springer 1995
H.M. Edwards: Riemann's Zeta Function. Academic Press 1974. Reprinted Dover
Hlawka / Schoißengeier / Taschner: Geometric and Analytic Number Theory. Springer 1991
A. Ivic: The Riemann Zeta-Function. Wiley 1985
S.J. Patterson: An introduction to the theory of the Riemann Zeta-Function. Cambridge UP 1988
K. Prachar: Prime number distribution. Springer 1957.
E.C. Titchmarsh: The Theory of the Riemann Zeta Function. Oxford UP, 2nd ed. 1986

Otto Forster 2008-10-20