: Mathematics: multiple zeta function

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multiple zeta function

In mathematics, the multiple zeta functions are generalizations of the Riemann zeta function, defined by
${\\displaystyle \\zeta (s_{1},\\ldots ,s_{k})=\\sum _{n_{1}>n_{2}>\\cdots >n_{k}>0}\\ {\ rac {1}{n_{1}^{s_{1}}\\cdots n_{k}^{s_{k}}}}=\\sum _{n_{1}>n_{2}>\\cdots >n_{k}>0}\\ \\prod _{i=1}^{k}{\ rac {1}{n_{i}^{s_{i}}}},\\!}$
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. The k in the above definition is named the "depth" of a MZV, and the n = s₁ + ... + s_k is known as the "weight".
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Multiple_zeta_function)

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