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
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(1)
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(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|)).](/images/equations/MultivariateZetaFunction/NumberedEquation2.svg) |
(2)
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(Broadhurst 1996; Bailey et al. 2007, p. 38).
The notation 
(as opposed to 
) is sometimes also used to indicate that a factor of 1
in the numerator is replaced by a corresponding factor of 
. In addition, the notation 
is used in quantum field theory.
In particular, for 
, these correspond to the usual Euler
sums
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(3)
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(4)
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(5)
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(6)
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(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
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(7)
|
for 
,
as well as
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(8)
|
for nonnegative integers 
and 
(Bailey et al. 2007). These give the special cases
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(9)
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(10)
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(11)
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(Bailey et al. 2007).
A different kind of special case is given by
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(12)
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(Borwein and Bailey 2003, p. 26; Borwein et al. 2004, Ch. 2, Ex. 29).
Other special values include
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(13)
|
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(14)
|
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(15)
|
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(16)
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(17)
|
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(18)
|
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(19)
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(Bailey et al. 2007, pp. 223 and 251). Closed forms are known for all 
with 
are known (Bailey et al. 2006, p. 39).
Amazingly,
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(20)
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found by J. Borwein and D. Broadhurst in 1996 (Bailey et al. 2006, p. 17).
-Fold Euler Sums and Their Roles in Knot Theory and Field Theory."
April 22, 1996. 
Primitives of Algebras of the Sixth Root of Unity."
March 11, 1998.