In 1986, Don Zagier generalized Euler's theorem ($\zeta_\mathbb{Q}(2)=\pi ^2 /6$) to an arbitrary number field $K$:
$$\zeta_K(2)=\frac{\pi^{2r+2s}}{\sqrt{|D|}}\times \sum_v c_v A(x_{v,1})…A(x_{v,s})$$
where
$$A(x)=\int^x_0 \frac{1}{1+t^2}\log \frac{4}{1+t^2}dt$$
In this sense, the result is conjectured to hold for $2<s\in \mathbb{N}$, with the $A(x)$ replaced by more complicated functions $A_m(x)$
This might seem rather unenlightening, but we can also state Zagier's result like this:
$$\zeta_K(2)=\text{the volume of a hyperbolic manifold}$$
This amazing fact doesn't seem to have a direct analogue for $\zeta_K(2m)$ with $m \neq 1$
- What I'd like to know is if there is any big picture explanation for the appearance of hyperbolic manifolds in this context.
Zagier's calculation is quite geometrical, but as far as I understand gives no clear explanation of "what the manifold is doing here".
Don Zagier, Hyperbolic manifolds and special values of Dedekind zeta-functions (1986)
Best Answer
As ThiKu mentions, the connection between $\zeta_K(2)$ and hyperbolic manifolds is that the volume formula for arithmetic hyperbolic manifolds is given by an explicit formula involving $\zeta_K(2)$. This formula is due to Borel in the case of arithmetic manifolds arising from quaternion algebras. A more general formula was later given by Prasad.
In order to get some intuition for general hyperbolic surfaces or $3$-manifolds, it might help to start by recalling the modular surface $\mathrm{SL}_2(\mathbb Z)\backslash \mathfrak h_2$ where $\mathfrak h_2$ denotes the hyperbolic plane. It is known that $\mathrm{vol}(\mathrm{SL}_2(\mathbb Z)\backslash \mathfrak h_2)=\frac{2}{\pi}\zeta(2)=\frac{\pi}{3}$. (See for instance Section 2 of these notes by Garret or these notes by Venkatesh.) So in this case we see the appearance of a zeta value in the context of the volume of a hyperbolic surface.
Here is an admittedly circuitous way to construct the hyperbolic surface $\mathrm{SL}_2(\mathbb Z)\backslash \mathfrak h_2$ that will provide some intuition for what happens more generally. Consider the quaternion algebra $B=\mathrm{M}_2(\mathbb Q)$ and the maximal order $\mathcal O=\mathrm{M}_2(\mathbb Z)$ inside this quaternion algebra. Let $\mathcal O^1=\mathrm{SL}_2(\mathbb Z)$ denote the multiplicative group of elements of $\mathcal O$ having determinant $1$ and let $\Gamma$ denote the image of $\mathcal O^1$ inside $\mathrm{SL}_2(\mathbb R)$ where we make use of the map $B\hookrightarrow B\otimes_{\mathbb Q} \mathbb R\cong \mathrm{M}_2(\mathbb R)$. The group $\Gamma$ is a discrete subgroup of $\mathrm{SL}_2(\mathbb R)$ which has finite volume, and as $\mathrm{SL}_2(\mathbb R)$ is the group of orientation preserving isometries of $\mathfrak h_2$, we get our quotient surface $\mathrm{SL}_2(\mathbb Z)\backslash \mathfrak h_2$.
The connection which Zagier makes between zeta values and manifolds ultimately arises from a "number field" analog of the above.
(A very interesting fact that is not directly related to your question by which I cannot resist mentioning is that many aspects of the geometry of manifolds defined in this sort of arithmetic manner (i.e., via the construction given below) are directly related to quantities of number theoretic interest. For instance, the lengths of closed geodesics on $\mathrm{SL}_2(\mathbb Z)\backslash \mathfrak h_2$ correspond to regulators of real quadratic fields and their multiplicity to the class number of the associated real quadratic field. This connection generalizes to manifolds constructed from number fields other than $\mathbb Q$ as well.)
Let $K$ be a number field with $r_1$ real places and $r_2$ complex places and $B$ be a quaternion algebra over $K$ which is not totally definite. Let $s$ denote the number of real places of $K$ that split in $B$. Thus $$B\otimes_{\mathbb Q} \mathbb R\cong \mathbb H^{r_1-s}\times \mathrm{M}_2(\mathbb R)^{s} \times \mathrm{M}_2(\mathbb C)^{r_2}.$$ Let $\mathcal O$ be a maximal order of $B$ and $\mathcal O^1$ the multiplicative subgroup consisting of elements with reduced norm $1$. (If $B$ was a matrix algebra then the reduced norm and determinant would coincide.) Let $\Gamma_\mathcal{O}$ denote the image of $\mathcal{O}^1$ in the group $$G_{s,r_2}=\mathrm{SL}_2(\mathbb R)^{s}\times \mathrm{SL}_2(\mathbb C)^{r_2}.$$ The group $G_{s,r_2}$ is the group of orientation preserving isometries of $\mathfrak h_2^{s}\times \mathfrak h_3^{r_2}$ which preserves factors. Here $\mathfrak h_3$ is hyperbolic $3$-space. The group $\Gamma_\mathcal O$ is a discrete subgroup of $G_{s,r_2}$ which is cocompact if $B$ is a division algebra and has covolume given by the formula $$\mathrm{vol}(\Gamma_\mathcal O\backslash \mathfrak h_2^{s}\times \mathfrak h_3^{r_2})=\frac{2(4\pi)^sd_K^{3/2}\zeta_K(2)}{(4\pi^2)^{r_1}(8\pi^2)^{r_2}}\prod_{\mathfrak p\in\mathrm{Ram}_f(B)}\left(N(\mathfrak p)-1\right),$$ where $d_K$ is the absolute value of the discriminant of $K$ and $\mathrm{Ram}_f(B)$ is the set of finite primes of $K$ which ramify in $B$. This formula is proven in Section 7.3 of Borel's paper. In particular note the appearance of $\zeta_K(2)$ in the formula.
The following two special cases of all of this are worth noting:
Finally, note that more general zeta values like $\zeta_K(2m)$ do appear in the volume formulas for different types of arithmetic manifolds. This is already the case for higher dimensional arithmetic hyperbolic manifolds. See for instance some of the formulas in this paper by Belolipetsky and Emery. (As was mentioned above, these zeta values arise when one works out the relevant case of Prasad's general volume formula.)