Multiple zeta function

In mathematics, the multiple zeta functions are generalizations of the Riemann zeta function, defined by

${\displaystyle \zeta (s_{1},\ldots ,s_{k})=\sum _{n_{1}>n_{2}>\cdots >n_{k}>0}\ {\frac {1}{n_{1}^{s_{1}}\cdots n_{k}^{s_{k}}}}=\sum _{n_{1}>n_{2}>\cdots >n_{k}>0}\ \prod _{i=1}^{k}{\frac {1}{n_{i}^{s_{i}}}},\!}$

and converge when Re(s₁) + ... + Re(s_i) > i for all i. Like the Riemann zeta function, the multiple zeta functions can be analytically continued to be meromorphic functions (see, for example, Zhao (1999)). When s₁, ..., s_k are all positive integers (with s₁ > 1) these sums are often called multiple zeta values (MZVs) or Euler sums. These values can also be regarded as special values of the multiple polylogarithms.^[1][2]

The k in the above definition is named the "depth" of a MZV, and the n = s₁ + ... + s_k is known as the "weight".^[3]

The standard shorthand for writing multiple zeta functions is to place repeating strings of the argument within braces and use a superscript to indicate the number of repetitions. For example,

${\displaystyle \zeta (2,1,2,1,3)=\zeta (\{2,1\}^{2},3).}$