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 is not expected have any 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)

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)

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 is not expected have any 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".

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 is not expected have any 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)