Ivan Shishkin, Rye (1878)

Problems/Functional analysisUnreviewed

Accessible values of the Riemann Zeta function

by Cypress·
92
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We define the Riemann Zeta function as follows: ∀s∈C, ℜ(s)>1, ζ(s)=∑k=1∞1ks\forall s\in \mathbb{C}, \: \Re(s) > 1, \:\zeta (s)= \sum_{k=1}^{\infty}\frac{1}{k^{s}}

Let k∈N∗.k \in \mathbb{N}^{*}. Show that the ζ(2k)\zeta(2k) can be expressed in terms of π2k\pi^{2k} and rational numbers, and provide, if possible an explicit formula.

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