Volumes of spheres and special values of zeta functions of and \mathbb{Z}/n\mathbb{Z}
Volume 208 / 2023
Abstract
The volume of the unit sphere in every dimension is given an 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 for the Riemann zeta function.