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Michel Balazard

An introduction to the zeta function and the Riemann hypothesis

Exercise sheets (in Russian!)


[μΙΣΤΟΛ 1 (32K)|μΙΣΤΟΛ 2 (28K)]

Zipped postscript

[μΙΣΤΟΛ 1 (14K)|μΙΣΤΟΛ 2 (13K)]

The conjecture, (RH), made by Riemann in 1859, that all nontrivial zeros of ζ(s) lie on the line Re(s)=1/2, is still open. I will give an introduction to the theory of the Riemann zeta function, with an emphasis on facts relevant to (RH), in particular equivalent formulations. Generalizations to Dedekind's zeta function and the Selberg class will be considered. The following topics will be discussed :

  1. Mellin transforms
  2. The functional equation
  3. Hardy's theorem (ζ(s) has an infinity of zeros on the line Re(s)=1/2)
  4. Hamburger's theorem (charcterization of ζ(s) by its functional equation)
  5. Numerical verification of (RH) (Turing's idea)
  6. Zeros and primes: the explicit formulas
  7. Weil's positivity criterion
  8. The criteria of Nyman and Baez-Duarte (reformulation of (RH) as an approximation problem in an Hilbert space)

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