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

Anders Karlsson; Massimiliano Pallich

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

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arxiv 2209.03590 v1 pith:G6X2O25D submitted 2022-09-08 math.NT

Anders Karlsson , Massimiliano Pallich This is my paper · ORCID

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keywords zetamathbbvaluesfunctionproductspecialvolumeakin

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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.

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