Desingularization of complex multiple zeta-functions
Desingularization of complex multiple zeta-functions
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DOI：
10.1353/ajm.2017.0002
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发表时间：
2015-08
影响因子：
1.7
通讯作者：
H. Furusho;Y. Komori;Kohji Matsumoto;Hirofumi Tsumura
中科院分区：
文献类型：
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作者：
H. Furusho;Y. Komori;Kohji Matsumoto;Hirofumi Tsumura
我们引入了多变量多重 zeta 函数（广义 Euler-Zagier 类型）的去奇异化方法，其动机是寻找非正整数点处多重 zeta 函数值的合适严格含义。我们发现，在进行去奇异化之后，多个 zeta 函数（已知在具有无限多个奇异超平面的整个空间中是亚纯的）在整个空间上是完整的。去奇异化函数由多个 zeta 函数的适当有限“线性”组合给出，其中一些参数发生了偏移。结果表明，伯努利数的特定组合在其非奇异化数的非正整数处获得特殊值。我们还讨论了扭曲的多重 zeta 函数，它们可以延续到整个函数，并且可以显式计算它们在非正整数点的特殊值。
We introduce the method of desingularization of multi-variable multiple zeta-functions (of the generalized Euler-Zagier type), under the motivation of finding a suitable rigorous meaning of the values of multiple zeta-functions at non-positive integer points. We reveal that multiple zeta-functions (which are known to be meromorphic in the whole space with infinitely many singular hyperplanes) turn out to be entire on the whole space after taking the desingularization. The desingularized function is given by a suitable finite "linear" combination of multiple zeta-functions with some arguments shifted. It is shown that specific combinations of Bernoulli numbers attain the special values at their non-positive integers of the desingularized ones. We also discuss twisted multiple zeta-functions, which can be continued to entire functions, and their special values at non-positive integer points can be explicitly calculated.