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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