[论文解读] Desingularization of complex multiple zeta-functions, fundamentals of $p$-adic multiple $L$-functions, and evaluation of their special values

Hidekazu Furusho, Yasushi Komori+2 more|arXiv (Cornell University)|Sep 16, 2013

Advanced Mathematical Identities参考文献 26被引用 3

一句话总结

本文引入了复多重zeta函数的去奇异化处理，通过平移zeta函数的线性组合使其成为整函数，并利用p进测度构造了p进多重L函数。关键结果包括：通过扭多重伯努利数显式计算p进多重L函数在非正整数处的取值，以及通过p进扭多重多对数函数在正整数处的取值，建立了多重库默尔同余关系，并在奇偶性条件下得到函数关系。

ABSTRACT

This paper deals with a multiple version of zeta- and L-functions both in the complex case and in the p-adic case: [I] Our motivation in the complex case is to find suitable rigorous meaning of the values of multivariable multiple zeta-functions (MZFs) at non-positive integer points. (a) We reveal that MZFs turn to be entire on the whole space after taking the desingularization. Further we show that the desingularized function is given by a suitable finite linear combination of MZFs with some arguments shifted. It is also shown that specific combinations of Bernoulli numbers attain the special values at their non-positive integers of the desingularized ones. (b) Twisted MZFs can be continued to entire functions and their special values at non-positive integer points can be explicitly calculated. [II] Our work in the p-adic case is to develop the study on analytic side of the Kubota-Leopoldt p-adic L-functions (pLFs) into the multiple setting. We construct p-adic multiple L-functions (pMLFs), multivariable versions of their pLFs, by using a specific p-adic measure. We establish their various fundamental properties: (a) We establish their intimate connection with the above complex MZFs by showing that the special values of pMLFs at non-positive integers are expressed by the twisted multiple Bernoulli numbers, the special values of the complex MZFs at non-positive integers. (b) We extend Kummer congruence for Bernoulli numbers to congruences for the twisted multiple Bernoulli numbers. (c) We extend the vanishing property of the Kubota-Leopoldt pLFs with odd characters to our pMLFs. (d) We establish their close relationship with the p-adic twisted multiple polylogarithms (pTMPLs) by showing that the special values of pMLFs at positive integers are described by those of pTMPLs at roots of unity, which generalizes the previous result of Coleman in the single variable case.

研究动机与目标

• 通过引入去奇异化过程，解决复多重zeta函数在非正整数处的不确定性问题。
• 通过p进测度将库博塔-莱普利希特p进L函数理论推广至多变量情形。
• 显式计算p进多重L函数在非正整数和正整数处的特殊值。
• 将库默尔同余关系与函数关系推广至多重p进L函数情形。
• 建立p进多重L函数与p进扭多重多对数函数在单位根处的联系。

提出的方法

• 将广义欧拉-扎吉耶r重zeta函数定义为平移zeta函数的有限线性组合，以消除奇点。
• 利用特定的p进测度构造p进多重L函数，确保其具有解析延拓与解析性质。
• 建立p进多重L函数在非正整数处的取值与扭多重伯努利数之间的联系。
• 通过p进插值法推导扭多重伯努利数的多重库默尔同余关系。
• 证明具有奇偶性条件的函数关系，推广奇特征p进L函数的零化性质。
• 将p进多重L函数在正整数处的特殊值与在单位根处取值的p进扭多重多对数函数联系起来。

实验结果

研究问题

• RQ1 尽管存在奇点，如何严格定义复多重zeta函数在非正整数处的取值？
• RQ2 p进多重L函数的结构是怎样的？它们如何推广经典p进L函数？
• RQ3 p进多重L函数在非正整数处的特殊值如何与扭多重伯努利数相关？
• RQ4 库默尔同余关系能否推广至多重p进L函数情形？
• RQ5 p进多重L函数与p进扭多重多对数函数在单位根处有何关系？

主要发现

• 通过平移zeta函数的有限线性组合，去奇异化的多重zeta函数在$\mathbb{C}^r$上成为整函数。
• 去奇异化zeta函数在非正整数处的特殊值由伯努利数的特定组合显式给出。
• 扭多重zeta函数是整函数，其在非正整数处的特殊值可显式计算。
• p进多重L函数在非正整数处的特殊值可用扭多重伯努利数表示，并与去奇异化的复zeta函数的特殊值相关联。
• 建立了扭多重伯努利数的多重库默尔同余关系，推广了经典库默尔同余关系。
• p进多重L函数在正整数处的特殊值可表示为在单位根处取值的p进扭多重多对数函数的线性组合，推广了单变量情形下科尔曼的结果。