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# Michel Balazard

## An introduction to the zeta function and the Riemann hypothesis

### Exercise sheets (in Russian!)

### Postscript

[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 :

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