Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 123 / 2 / 1

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- Solution 1

Solution

For

Re s > 1

, the Dedekind zeta function of

K

is

ζ_{K} (s) = \sum_{0 \neq = a \subseteq O_{K}} (N a)^{- s} .

Unique factorization of ideals gives its Euler product

ζ_{K} (s) = \prod_{p} (1 - (N p)^{- s})^{- 1},

over the nonzero prime ideals of

O_{K}

.

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