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Multivariate Zeta Function


Multivariate zeta function, also called multiple zeta values, multivariate zeta constants (Bailey et al. 2006, p. 43), multi-zeta values (Bailey et al. 2006, p. 17), and multivariate zeta values, are defined by

 zeta(s_1,...,s_k; sigma_1,...,sigma_k)=sum_(n_1>n_2>...>n_k>0)product_(j=1)^k(sigma_j^(n_j))/(n_j^(s_j))
(1)

(Broadhurst 1996, 1998). This can be written in the more compact and convenient form

 zeta(a_1,...,a_k)=sum_(n_1>n_2>...>n_k>0)product_(j=1)^k([sgn(a_j)]^(n_j))/(n_j^(|a_j|)) 
=sum_(n_1>n_2>...>n_k>0)([sgn(a_1)]^(n_1)[sgn(a_2)]^(n_2)...[sgn(a_k)]^(n_k))/(n_1^(|a_1|)n_2^(|a_2|)...|n_k|^(|a_k|)).
(2)

(Broadhurst 1996; Bailey et al. 2007, p. 38).

The notation a^__k (as opposed to -a_k) is sometimes also used to indicate that a factor of 1 in the numerator is replaced by a corresponding factor of (-1)^(n_k). In addition, the notation U(s,t)=zeta(-t,s) is used in quantum field theory.

In particular, for k=2, these correspond to the usual Euler sums

zeta(s,t)=sigma_h(t,s)
(3)
zeta(-s,t)=alpha(t,s)
(4)
zeta(s,-t)=-sigma_a(t,s)
(5)
zeta(-s,-t)=-alpha_a(t,s)
(6)

(Broadhurst 1996).

Multivariate zeta functions (and their derivatives) also arise in the closed-form evaluation of definite integrals involving the log cosine function (Oloa 2011).

These sums satisfy

 zeta(a,b)+zeta(b,a)=zeta(a)zeta(b)-zeta(a+b)
(7)

for a,b>1, as well as

 sum_(suma_i=n; a_i>=0)zeta(a_1+2,a_2+1,...,a_r+1)=zeta(n+r+1)
(8)

for nonnegative integers n and r (Bailey et al. 2007). These give the special cases

zeta(3)=zeta(2,1)
(9)
zeta(4)=zeta(3,1)+zeta(2,2)
(10)
zeta(2,1,1)=zeta(4)
(11)

(Bailey et al. 2007).

A different kind of special case is given by

 zeta(3,1_()_(n))=(2pi^(4n))/((4n+2)!)
(12)

(Borwein and Bailey 2003, p. 26; Borwein et al. 2004, Ch. 2, Ex. 29).

Other special values include

zeta(-3,-1)=-1/(12)(ln2)^4+1/2zeta(4)+1/2zeta(2)(ln2)^2-2Li_4(1/2)
(13)
zeta(-2,-1)=3/2zeta(2)ln2-(13)/8zeta(3)
(14)
zeta(-2,1)=1/8zeta(3)
(15)
zeta(2,-1)=zeta(3)-3/2zeta(2)ln2
(16)
zeta(2,1)=zeta(3)
(17)
zeta(2,1,1)=zeta(4)
(18)
zeta(3,1)=1/(360)pi^4
(19)

(Bailey et al. 2007, pp. 223 and 251). Closed forms are known for all zeta(a_1,...,a_k) with sum_(k)|a_k|<8 are known (Bailey et al. 2006, p. 39).

Amazingly,

 zeta(2,1_()_(n))=8^nzeta(-2,1_()_(n)),
(20)

found by J. Borwein and D. Broadhurst in 1996 (Bailey et al. 2006, p. 17).


See also

Euler Sum, Log Sine Function, Multiple Series, Tetrahedral Vacuum Feynman Diagram

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References

Akiyama, S.; Egami, S.; and Tanigawa, Y. "Analytic Continuation of Multiple Zeta-Functions and Their Values at Non-Positive Integers." Acta Arith. 98, 107-116, 2001.Bailey, D. H.; Borwein, J. M.; Calkin, N. J.; Girgensohn, R.; Luke, D. R.; and Moll, V. H. "Computation of Multivariate Zeta Constants." §2.5 in Experimental Mathematics in Action. Wellesley, MA: A K Peters, pp. 43 and 223-224, 2007.Bailey, D. H.; Borwein, J. M.; Kapoor, V.; and Weisstein, E. W. "Ten Problems in Experimental Mathematics." Amer. Math. Monthly 113, 481-509, 2006.Borwein, J. and Bailey, D. "Quantum Field Theory." §2.6 in Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, pp. 58-59, 2003.Borwein, J.; Bailey, D.; and Girgensohn, R. Ch. 3 in Experimentation in Mathematics: Computational Paths to Discovery. Wellesley, MA: A K Peters, 2004.Borwein, J. M.; Bradley, D. M.; Broadhurst, D. J.; and Lisonek, P. "Special Values of Multidimensional Polylogarithms." Trans. Amer. Math. Soc. 353, 907-941, 2001.Broadhurst, D. J. "On the Enumeration of Irreducible k-Fold Euler Sums and Their Roles in Knot Theory and Field Theory." April 22, 1996. http://arxiv.org/abs/hep-th/9604128Broadhurst, D. J. "Massive 3-Loop Feynman Diagrams Reducible to SC^* Primitives of Algebras of the Sixth Root of Unity." March 11, 1998. http://arxiv.org/abs/hep-th/9803091.Oloa, O. "A Log-Cosine Integral Involving a Derivative of a MZV." Preprint. Apr. 18, 2011.

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Multivariate Zeta Function

Cite this as:

Weisstein, Eric W. "Multivariate Zeta Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MultivariateZetaFunction.html

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