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

TL;DR

This paper links the volume of high-dimensional spheres to special values of zeta functions, providing new formulas and insights into classical zeta values, including Euler's and Catalan numbers.

Contribution

It introduces a novel interpretation of sphere volumes via zeta function values and derives new product formulas and closed-form expressions for these values.

Findings

Volume of spheres expressed as zeta function products

Derived product formula for zeta functions related to Catalan numbers

New perspectives on Euler's zeta values

Abstract

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.