Volumes of spheres and special values of zeta functions of $\mathbb{Z}$ and $\mathbb{Z}/n\mathbb{Z}$
Volume 208 / 2023
Acta Arithmetica 208 (2023), 161-170
MSC: Primary 11M41; Secondary 28A75, 05A15.
DOI: 10.4064/aa220912-1-3
Published online: 28 March 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.
