[Paper Review] 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 Identities26 references3 citations

TL;DR

This paper introduces a desingularization of complex multiple zeta-functions, rendering them entire via linear combinations of shifted zeta-functions, and constructs $p$-adic multiple $L$-functions using $p$-adic measures. Key results include explicit evaluations of $p$-adic multiple $L$-functions at non-positive integers via twisted multiple Bernoulli numbers and at positive integers via $p$-adic twisted multiple polylogarithms, establishing multiple Kummer congruences and functional relations with parity conditions.

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.

Motivation & Objective

• To resolve the indeterminacy of complex multiple zeta-functions at non-positive integers by introducing a desingularization process.
• To extend the theory of Kubota-Leopoldt $p$-adic $L$-functions to multivariable settings via $p$-adic measures.
• To establish explicit evaluations of $p$-adic multiple $L$-functions at non-positive and positive integers.
• To generalize Kummer congruences and functional relations to the multiple $p$-adic $L$-function setting.
• To connect $p$-adic multiple $L$-functions to $p$-adic twisted multiple polylogarithms at roots of unity.

Proposed method

• Define a desingularization of generalized Euler-Zagier multiple zeta-functions as a finite linear combination of shifted zeta-functions to remove singularities.
• Construct $p$-adic multiple $L$-functions using a specific $p$-adic measure, ensuring meromorphic continuation and analyticity.
• Establish a connection between $p$-adic multiple $L$-function values at non-positive integers and twisted multiple Bernoulli numbers.
• Derive multiple Kummer congruences for twisted multiple Bernoulli numbers via $p$-adic interpolation.
• Prove functional relations with parity conditions, generalizing the vanishing property of odd-character $p$-adic $L$-functions.
• Relate positive integer special values of $p$-adic multiple $L$-functions to $p$-adic twisted multiple polylogarithms evaluated at roots of unity.

Experimental results

Research questions

• RQ1 How can the values of complex multiple zeta-functions at non-positive integers be rigorously defined despite their singularities?
• RQ2 What is the structure of $p$-adic multiple $L$-functions, and how do they generalize classical $p$-adic $L$-functions?
• RQ3 How are the special values of $p$-adic multiple $L$-functions at non-positive integers related to twisted multiple Bernoulli numbers?
• RQ4 Can Kummer congruences be extended to the multiple $p$-adic $L$-function setting?
• RQ5 What is the relationship between $p$-adic multiple $L$-functions and $p$-adic twisted multiple polylogarithms at roots of unity?

Key findings

• The desingularized multiple zeta-function becomes entire on $\mathbb{C}^r$ through a finite linear combination of shifted zeta-functions.
• Special values of the desingularized zeta-function at non-positive integers are given explicitly by specific combinations of Bernoulli numbers.
• Twisted multiple zeta-functions are entire and their special values at non-positive integers are explicitly computable.
• The special values of $p$-adic multiple $L$-functions at non-positive integers are expressed in terms of twisted multiple Bernoulli numbers and the special values of the desingularized complex zeta-function.
• Multiple Kummer congruences are established for twisted multiple Bernoulli numbers, generalizing the classical Kummer congruence.
• The special values of $p$-adic multiple $L$-functions at positive integers are described as linear combinations of $p$-adic twisted multiple polylogarithms evaluated at roots of unity, generalizing Coleman’s result in the single-variable case.