$F_1$格式中某些变形多变量zeta函数的亚纯性
Meromorphicity of some deformed multivariable zeta functions for $F_1$-schemes
摘要：
受Deitmar-Koyama-Kurokawa，Kurokawa-Ochiai，Connes-Consani和作者最近工作的启发，我们在Connes-Consani意义下定义了Noether $\F_1$ -格式$X$的Hurwitz-Igusa型多变量变形zeta函数.我们的zeta函数推广了Deitmar-Koyama-Kurokawa，Kurokawa-Ochiai研究的zeta函数，以及修正的Soulé型zeta函数Connes-Consani的对数导数。我们使用Hurwitz zeta函数给出了这些zeta函数的显式表示，因此，我们可以导出其亚纯性。当仅限于修正的Soulé型zeta函数的对数导数时，我们发现ArXiv-0907.0918v2中引入的有限阿贝尔群$A$的不变量$μ(A)$在Soulé型zeta函数中起着极其重要的作用.
英文摘要：
Motivated by recent work of Deitmar-Koyama-Kurokawa, Kurokawa-Ochiai, Connes-Consani, and the author, we define some multivariable deformed zeta functions of Hurwitz-Igusa type for a Noetherian $\F_1$-scheme $X$ in the sense of Connes-Consani. Our zeta functions generalize both the zeta functions studied by Deitmar-Koyama-Kurokawa, Kurokawa-Ochiai, and the log derivative of the modified Soulé type zeta function Connes-Consani. We give an explicit presentation for these zeta functions using the Hurwitz zeta functions, and so, we can derive its meromorphicity. When restricted to the log derivative of the modified Soulé type zeta functions, we find our invariant $μ(A)$ for a finite abelian group $A$, introduced in ArXiv-0907.0918v2, plays an extremely important role in the Soulé type zeta functions.
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