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