By giving the definition of the sum of a series indexed by a set on which a group operates, we prove that the sum of the series that defines the Riemann zeta function, the Epstein zeta function and a few other series indexed by \({\mathbb {Z}}^k\) has an intrinsic meaning as a complex number, independent of the requirements of analytic continuation. The definition of the sum requires nothing more than algebra and the concept of absolute convergence. The analytical significance of the algebraically defined sum is then explained by an argument that relies on the Poisson formula for tempered distributions.

中文翻译：

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求和与泊松公式

通过给出由一组在其上进行操作的集合索引的序列之和的定义，我们证明了定义Riemann zeta函数，Epstein zeta函数以及由\（{ mathbb {Z}} ^ k \）的固有含义是复数，与解析连续性的要求无关。和的定义只需要代数和绝对收敛的概念即可。然后，通过一个依赖于Poisson公式进行回火分布的论点来解释代数定义的总和的分析意义。