We examine “partition zeta functions” analogous to the Riemann zeta function but summed over subsets of integer partitions. We prove an explicit formula for a family of partition zeta functions already shown to have nice properties — those summed over partitions of fixed length — which yields complete information about analytic continuation, poles and trivial roots of the zeta functions in the family. Then we present a combinatorial proof of the explicit formula, which shows it to be a zeta function analog of MacMahon’s partial fraction decomposition of the generating function for partitions of fixed length.

中文翻译：

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分区zeta函数的分析与组合

我们检查类似于黎曼 zeta 函数的“分区 zeta 函数”，但对整数分区的子集求和。我们证明了一个已证明具有良好性质的分区 zeta 函数族的显式公式——那些对固定长度的分区求和的函数——它产生了关于该族 zeta 函数的解析延拓、极点和平凡根的完整信息。然后我们给出了显式公式的组合证明，表明它是 MacMahon 对固定长度分区的生成函数的部分分数分解的 zeta 函数模拟。