For Riemann's zeta function, one knows that:

$\zeta(2n)$ is irrational (because a rational multiple of $\pi^{2n}$ is)

$\zeta(3)$ is irrational (proved by Apéry)

and a few other results like "there are infinitely many irrational values at odd integers" (Ball-Rivoal).

Is there something known for the values of the Dedekind's zeta function $\zeta_K(s)$ of a number field? For instance, do we know the irrationality of $\zeta_K(2)$ in some cases?

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    $\begingroup$ The Klingen-Siegel theorem implies that if $K/\mathbb{Q}$ is totally real of degree $d$ and $s$ is an even positive integer, then $\zeta_{K}(s)$ is a rational multiple of $\frac{\pi^{d+s}}{\sqrt{D}}$, where $D$ is the discriminant of $K$, and is hence irrational (this time by the transcendence of $\pi$). $\endgroup$ – Jeremy Rouse Feb 25 '15 at 17:23
  • $\begingroup$ $\zeta(2n)$ is irrational because it is a rational times a power of $\pi$, and powers of $\pi$ are irrational --- but not because $\pi$ is irrational. $\endgroup$ – Gerry Myerson Feb 25 '15 at 22:09
  • $\begingroup$ Jeremy: Your statement of Klingen--Siegel seems to have some issues when $d = 1$ :-) Should it maybe be $\pi^{ds}$ in the denominator? $\endgroup$ – David Loeffler Feb 26 '15 at 7:41
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    $\begingroup$ David - You are correct. The power of $\pi$ in the numerator should be $ds$. (I was copying the statement from Zagier, and Zagier uses the notation $n_{+}$ for $d$, hence the confusion.) $\endgroup$ – Jeremy Rouse Feb 26 '15 at 14:26

As with all motivic L-functions, irrationality of special values breaks into two rather different problems:

Critical values. In the case of Dedekind zeta functions, those are the even positive integers, and the strategy is to show that $\zeta_K(-2n-1)$ is rational, so that from the functional equation follows that $\zeta_K(2n)$ is irrational. Of course, this only works for totally real number fields, since that's the only case where the functional equation doesn't vanish at the odd integers.

This is precisely the Siegel-Klingen theorem, $\zeta_K(2n)$ is a rational multiple of $\pi^{2n[K:\mathbb{Q}]}$, and therefore irrational.

The actual value of that algebraic number is in general not known, see the Lichtenbaum conjectures.

Non-critical values. Odd positive integers. This is the (even more) difficult part, because you have to actually prove that something non-trivial is irrational. As you mention, some results are known for Riemann's zeta, but as far as I know, there are no known results for number fields other than $\mathbb{Q}$.

(On a side note, a generalized "infinitely many irrational numbers" type result for Dirichlet L-functions was published a few years ago by Masaki Nishimoto (paper here)).


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