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The volume of the unit sphere in every dimension is given a newinterpretation as a product of special values of the zeta function of$\mathbb{Z}$, akin to volume formulas of Minkowski and Siegel in the theory ofarithmetic groups. A product formula is found for this zeta function thatspecializes to Catalan numbers. Moreover, certain closed-form expressions forvarious other zeta values are deduced, in particular leading to an alternativeperspective on Euler's values of the Riemann zeta function.

Volumes of spheres and special values of zeta functions of $\mathbb{Z}$ and $\mathbb{Z}/n\mathbb{Z}$