**The arithmetic of special values of the Riemann zeta function**

The Riemann zeta function

ζ(*s*) = 1^{-s} + 2^{-s}
+ 3^{-s} + ...
= Σ_{n≥1} *n*^{-s}
is of fundamental importance in arithmetic and is the subject of many
deep conjectures. For example, the famous Riemann conjecture asserts
that all its non-trivial zeroes (i.e. the ones that are not real)
have real part 1/2. In this talk I want to explain the arithmetic
significance of the other zeroes and more generally of the values
of ζ(*n*) for all integers *n*.
This kind of L-function can be associated to every algebriac variety,
and a very general and deep conjecture due to A. Beilinson and
S. Bloch/K. Kato predicts their values at integers in terms
of the arithmetic of the variety in question. It is possible to verify
this conjecture for ζ(*s*).

In the last part of my talk I will survey what is known about the
general conjecture.

*Guido Kings*

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