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Special values of Dedekind zeta functions can frequently be expressed (and then calculated) in topological terms, e.g. from the Euler characteristics of certain manifolds/orbifolds.

An easy-to-formulate example is that of zeta functions of totally real quadratic fields $K$: the value of $\zeta_K(2)$ can be computed from the orbifold Euler characteristics of the Hilbert modular surface $$SL(2,O_K)\backslash (H^2\times H^2),$$ where $SL_2(O_K)$ is embedded into $SL(2,R)\times SL(2,R)=Isom^+(H^2\times H^2)$ via the two different embeddings $K\to R$.

More involved examples can be found in http://people.mpim-bonn.mpg.de/zagier/files/scanned/ValuesofZetaFunsAndApplications/fulltext.pdf and its references.

In a similar vein, the class number of $O_K$ equals the number of cusps (= the number of ends) of the Hilbert modular surface, which (I hope) might be easier to compute.

Special values of Dedekind zeta functions can frequently be expressed (and then calculated) in topological terms, e.g. from the Euler characteristics of certain manifolds/orbifolds.

An easy-to-formulate example is that of zeta functions of totally real quadratic fields $K$: the value of $\zeta_K(2)$ can be computed from the orbifold Euler characteristics of the Hilbert modular surface $$SL(2,O_K)\backslash (H^2\times H^2),$$ where $SL_2(O_K)$ is embedded into $SL(2,R)\times SL(2,R)\subset Isom^+(H^2\times H^2)$ via the two different embeddings $K\to R$.

More involved examples can be found in http://people.mpim-bonn.mpg.de/zagier/files/scanned/ValuesofZetaFunsAndApplications/fulltext.pdf and its references.

In a similar vein, the class number of $O_K$ equals the number of cusps (= the number of ends) of the Hilbert modular surface, which (I hope) might be easier to compute.

Special values of Dedekind zeta functions can frequently be expressed (and then calculated) in topological terms, e.g. from the Euler characteristics of certain manifolds/orbifolds.

An easy-to-formulate example is that of zeta functions of totally real quadratic fields $K$: the value of $\zeta_K(2)$ can be computed from the orbifold Euler characteristics of the Hilbert modular surface $$SL(2,O_K)\backslash (H^2\times H^2),$$ where $SL_2(O_K)$ is embedded into $SL(2,R)\times SL(2,R)=Isom^+(H^2\times H^2)$ via the two different embeddings $K\to R$.

More involved examples can be found in http://people.mpim-bonn.mpg.de/zagier/files/scanned/ValuesofZetaFunsAndApplications/fulltext.pdf and its references.

In a similar vein, the class number of $O_K$ is the number of cusps (= the number of ends) of the Hilbert modular surface, which (I hope) might be easier to compute.

Special values of Dedekind zeta functions can frequently be expressed (and then calculated) in topological terms, e.g. from the Euler characteristics of certain manifolds/orbifolds.

An easy-to-formulate example is that of zeta functions of totally real quadratic fields $K$: the value of $\zeta_K(2)$ can be computed from the orbifold Euler characteristics of the Hilbert modular surface $$SL(2,O_K)\backslash (H^2\times H^2),$$ where $SL_2(O_K)$ is embedded into $SL(2,R)\times SL(2,R)=Isom^+(H^2\times H^2)$ via the two different embeddings $K\to R$.

More involved examples can be found in http://people.mpim-bonn.mpg.de/zagier/files/scanned/ValuesofZetaFunsAndApplications/fulltext.pdf and its references.

In a similar vein, the class number of $O_K$ equals the number of cusps (= the number of ends) of the Hilbert modular surface, which (I hope) might be easier to compute.

Special values of Dedekind zeta functions can frequently be expressed (and then calculated) in topological terms, e.g. from the Euler characteristics of certain manifolds/orbifolds.

An easy-to-formulate example is that of zeta functions of totally real quadratic fields $K$: the value of $\zeta_K(2)$ can be computed from the orbifold Euler characteristics of the Hilbert modular surface $$SL(2,O_K)\backslash (H^2\times H^2),$$ where $SL_2(O_K)$ is embedded into $SL(2,R)\times SL(2,R)=Isom^+(H^2\times H^2)$ via the two different embeddings $K\to R$.

More involved examples can be found in http://people.mpim-bonn.mpg.de/zagier/files/scanned/ValuesofZetaFunsAndApplications/fulltext.pdf and its references.

In a similar vein, the class number of $O_K$ is the number of cusps of the Hilbert modular surface, which (I hope) might be easier to compute.

Special values of Dedekind zeta functions can frequently be expressed (and then calculated) in topological terms, e.g. from the Euler characteristics of certain manifolds/orbifolds.

An easy-to-formulate example is that of zeta functions of totally real quadratic fields $K$: the value of $\zeta_K(2)$ can be computed from the orbifold Euler characteristics of the Hilbert modular surface $$SL(2,O_K)\backslash (H^2\times H^2),$$ where $SL_2(O_K)$ is embedded into $SL(2,R)\times SL(2,R)=Isom^+(H^2\times H^2)$ via the two different embeddings $K\to R$.

More involved examples can be found in http://people.mpim-bonn.mpg.de/zagier/files/scanned/ValuesofZetaFunsAndApplications/fulltext.pdf and its references.

In a similar vein, the class number of $O_K$ is the number of cusps (= the number of ends) of the Hilbert modular surface, which (I hope) might be easier to compute.