The arithmetic of special values of the Riemann zeta function
The Riemann zeta function
ζ(s) = 1-s + 2-s
+ 3-s + ...
= Σn≥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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