$\mathbb{Z}$和$\mathbb{Z}/n\mathbb{Z}$的球面体积和zeta函数的特殊值

IF 0.5 3区数学 Q3 MATHEMATICS

Acta Arithmetica Pub Date : 2022-09-08 DOI:10.4064/aa220912-1-3

A. Karlsson, Massimiliano Pallich

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引用次数: 0

摘要

将单位球在每一维上的体积解释为$\mathbb{Z}$的ζ函数的特殊值的乘积，类似于算术群理论中的Minkowski和Siegel的体积公式。为这个专门用于加泰罗尼亚数字的zeta函数找到了一个乘积公式。此外，还推导了各种其他zeta值的某些封闭形式表达式，特别是导致黎曼zeta函数的欧拉值的另一种观点。

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

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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