Multivariate zeta function, also called multiple zeta values, multivariate zeta constants , multi-zeta values , and multivariate zeta values, are defined by ζ(s_1, ..., s_k σ_1, ..., σ_k) congruent sum_(n_1>n_2>...>n_k>0) product_(j = 1)^k σ_j^(n_j)/n_j^(s_j) (Broadhurst 1996, 1998). This can be written in the more compact and convenient form ζ(a_1, ..., a_k) = sum_(n_1>n_2>...>n_k>0) product_(j = 1)^k [sgn(a_j)]^(n_j)/n_j^( left bracketing bar a_j right bracketing bar ) = 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^( left bracketing bar a_1 right bracketing bar ) n_2^( left bracketing bar a_2 right bracketing bar ) ...( left bracketing bar n_k right bracketing bar )^( left bracketing bar a_k right bracketing bar )).
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