[Paper Review] Desingularization of complex multiple zeta-functions

Hidekazu Furusho, Yasushi Komori+2 more|arXiv (Cornell University)|Aug 27, 2015

Advanced Mathematical Identities16 references3 citations

TL;DR

This paper introduces a desingularization method to resolve the infinitely many singular hyperplanes of complex multiple zeta-functions of the generalized Euler-Zagier type, transforming them into entire functions via finite linear combinations with shifted arguments. The key result is that the desingularized function yields well-defined special values at non-positive integers, explicitly expressed in terms of Bernoulli numbers.

ABSTRACT

We introduce the method of desingularization of multi-variable multiple zeta-functions (of the generalized Euler-Zagier type), under the motivation of finding 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 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.

Motivation & Objective

• To resolve the indeterminacy of multiple zeta-functions at non-positive integer points, which lie on infinitely many singular hyperplanes.
• To provide a rigorous, unambiguous definition of multiple zeta-function values at non-positive integers, overcoming the ambiguity of limit-dependent results.
• To generalize the idea of removing singularities (as in (s−1)ζ(s) being entire) to the multi-variable case.
• To extend the desingularization approach to twisted multiple zeta-functions and derive their entire continuation and special values.
• To establish explicit formulas for desingularized values in terms of Bernoulli numbers, particularly for the case of generalized Euler-Zagier type.

Proposed method

• Construct a desingularized multiple zeta-function as a finite linear combination of shifted multiple zeta-functions with coefficients depending on the variables.
• Use the structure of singular hyperplanes (e.g., s_r=1, s_{r-1}+s_r=2,1,0,...) to guide the selection of shifts and coefficients to cancel singularities.
• Apply the method to the generalized Euler-Zagier type multiple zeta-function ζ_r((s_j);(γ_j)) and its twisted version with roots of unity.
• Derive explicit formulas for the desingularized function at non-positive integers using multiple Bernoulli numbers.
• Verify the entire nature of the desingularized function by showing cancellation of all singularities through algebraic identities.
• Utilize known identities such as ζ_2(0,s) = ζ(s−1)−ζ(s) to relate the desingularized values to known zeta functions and validate results.

Experimental results

Research questions

• RQ1 Can a rigorous, unambiguous value be assigned to multiple zeta-functions at non-positive integer points despite their indeterminacy due to singular hyperplanes?
• RQ2 Is it possible to construct an entire function from the meromorphic multiple zeta-function by a finite linear combination of its shifted versions?
• RQ3 What is the explicit form of the special values of the desingularized multiple zeta-function at non-positive integers?
• RQ4 How do the special values of the desingularized function relate to multiple Bernoulli numbers?
• RQ5 Are there relationships between the desingularization method and existing renormalization techniques (e.g., Guo-Zhang, Manchon-Paycha)?

Key findings

• The desingularized multiple zeta-function is entire in C^r, with all singularities canceled by a finite linear combination of shifted zeta-functions.
• For r=2, the desingularized value at (−k,−l) is given by a double sum over Bernoulli numbers: ζ²_des(−k,−l;1,1) = (−1)^{k+l} Σ Σ Σ B_{k+ν+κ+1}B_{l−κ+ρ+1}B_{m−ν−ρ+1} γ₁^{k+ν+κ+1}γ₂^{l−κ+ρ+1} with multinomial coefficients.
• For r=3, the desingularized value at (−k,−l,−m) is explicitly computed as a triple sum over Bernoulli numbers and multinomial coefficients.
• The desingularized function at (−1,1;1,1) evaluates to 1/8, and at (−1,4;1,1) to ζ(3)−ζ(4), demonstrating consistency with known zeta values.
• The method yields distinct values from renormalization techniques: for example, ζ²_des(0,−2;1,1) = 1/18, differing from Guo-Zhang’s 1/120 and Manchon-Paycha’s 7/720.
• The desingularized function allows well-defined evaluation at non-admissible indices (e.g., (1,1), (2,1)), yielding ζ²_des(1,1;1,1) = 1/2 and ζ²_des(2,1;1,1) = −ζ(2)+2ζ(3).