[Paper Review] Sum formula for multiple zeta function

Minoru Hirose, Hideki Murahara+1 more|arXiv (Cornell University)|Aug 5, 2018

Advanced Mathematical Identities3 references3 citations

TL;DR

This paper generalizes the classical sum formula for multiple zeta values (MZVs) to the Euler-Zagier multiple zeta function (MZF) with complex arguments. It introduces a recursive family of functions $ G_{a,b}(s_1,\dots,s_a;s) $ that unify MZVs of higher depth, proving that $ G_{0,b}(s) = \zeta(s) $ for $ \Re(s) > b $, thus extending the sum formula to complex variables and depth $ b \geq 2 $.

ABSTRACT

The sum formula is a well known relation in the field of the multiple zeta values. In this paper, we present its generalization for the Euler-Zagier multiple zeta function.

Motivation & Objective

• To extend the classical sum formula for multiple zeta values (MZVs) from positive integers to complex arguments in the Euler-Zagier multiple zeta function (MZF).
• To resolve Matsumoto's question on whether known MZV relations hold beyond positive integers by generalizing the sum formula to complex variables.
• To define and analyze a recursive family of functions $ G_{a,b}(s_1,\dots,s_a;s) $ that unify MZVs of depth $ a+b $, ensuring convergence under specified complex domain conditions.
• To prove that $ G_{0,b}(s) = \zeta(s) $ for $ \Re(s) > b $, thereby recovering the classical sum formula as a special case when $ s \in \mathbb{Z}_{>b} $.

Proposed method

• Define $ G_{a,b}(s_1,\dots,s_a;s) $ recursively via infinite series involving MZF values: $ G_{a,b} = \sum_{n=0}^\infty G_{a+1,b-1}(s_1,\dots,s_a,s-n-b;n+b) - \sum_{n=0}^\infty G_{a+1,b-1}(s_1,\dots,s_a,-n;s+n) $, with base case $ G_{a,1} = \zeta(s_1,\dots,s_a,s) $.
• Establish absolute convergence of $ G_{a,b} $ under the domain conditions $ \Re(s) > b $, $ \Re(s + s_a) > 1 + b $, ..., $ \Re(s + s_a + \cdots + s_1) > a + b $, using comparison with multiple zeta series.
• Prove an explicit integral representation of $ G_{a,b} $ as a sum over chains $ 0 < m_1 < \cdots < m_a < m $, with coefficients involving harmonic sums over $ x_1 \leq \cdots \leq x_{b-1} $.
• Use analytic continuation to extend the identity $ \sum_{n=0}^\infty (\zeta(s-n-2,n+2) - \zeta(-n,s+n)) = \zeta(s) $ from $ \Re(s) > 2 $ to $ \Re(s) > 1 $, excluding $ s = 2 $.
• Leverage recursive identities involving $ F_d^{(i)}(D;s) $, which represent partial sums over chains of integers, to prove $ F_{d+1}(D;s) = F_d^{(1)}(D;s) - F_d^{(2)}(D;s) - F_d^{(3)}(D;s) $, enabling inductive convergence control.
• Apply the recursive structure to show that $ G_{0,b}(s) = \zeta(s) $ by reducing the expression to a sum over $ m $ with harmonic mean terms, ultimately yielding the Riemann zeta function.

Experimental results

Research questions

• RQ1 Can the classical sum formula for MZVs be generalized to complex arguments in the Euler-Zagier MZF?
• RQ2 What recursive structure governs the sum of MZVs of depth $ b $ with complex parameters, ensuring convergence and analyticity?
• RQ3 How does the generalized sum formula relate to the classical sum formula when $ s $ is a positive integer greater than $ b $?
• RQ4 What conditions on complex parameters ensure the convergence and meromorphic continuation of the generalized sum formula?
• RQ5 Can the sum formula be extended to arbitrary depth $ b \geq 2 $ using recursive families of MZF values?

Key findings

• The generalized sum formula $ G_{0,b}(s) = \zeta(s) $ holds for all complex $ s $ with $ \Re(s) > b $, extending the classical sum formula to complex parameters.
• The function $ G_{a,b}(s_1,\dots,s_a;s) $ is well-defined and absolutely convergent under the domain conditions $ \Re(s) > b $, $ \Re(s + s_a) > 1 + b $, ..., $ \Re(s + s_a + \cdots + s_1) > a + b $.
• For $ s \in \mathbb{Z}_{>b} $, $ G_{a,b}(s_1,\dots,s_a;s) $ reduces to the sum of MZVs $ \sum_{m_1+\cdots+m_b=s,\, m_i \geq 1, m_b \geq 2} \zeta(s_1,\dots,s_a,m_1,\dots,m_b) $, recovering the classical sum formula.
• The identity $ \sum_{n=0}^\infty (\zeta(s-n-2,n+2) - \zeta(-n,s+n)) = \zeta(s) $ is proven for $ \Re(s) > 2 $ via series transformation and extended to $ \Re(s) > 1 $, $ s \neq 2 $, by analytic continuation.
• The recursive structure of $ G_{a,b} $ is shown to satisfy $ F_{d+1}(D;s) = F_d^{(1)}(D;s) - F_d^{(2)}(D;s) - F_d^{(3)}(D;s) $, enabling inductive convergence control and explicit evaluation.
• The explicit formula $ G_{a,b}(s_1,\dots,s_a;s) = \sum_{0<m_1<\cdots<m_a<m} \frac{1}{m_1^{s_1}\cdots m_a^{s_a} m^{s-b+1}} \sum_{m-m_a \leq x_1 \leq \cdots \leq x_{b-1} \leq m} \frac{1}{x_1 \cdots x_{b-1}} $ is derived and proven convergent.