Volumes of spheres and special values of zeta functions of ℤ and ℤ/nℤ

2022/09/08 by Anders Karlsson, Karlsson, Anders, Massimiliano Pallich +1
Mathematics · Physics and Astronomy · #11M41 #28A75 #Advanced Mathematical Identities #Advanced Mathematical Theories and Applications #FOS: Mathematics #History and Theory of Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2209.03590

openalex publication_date 2022/09/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 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.

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