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Multiple zeta function

In mathematics, the multiple zeta functions are generalizations of the Riemann zeta function, defined by : \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(s 1 ) + ... + 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 s 1 , ..., s k are all positive integers (with s 1 > 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 = s 1 + ... + s k is known as the "weight". 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, : \zeta(2,1,2,1,3) = \zeta(\{2,1\}^2,3).

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Article summary

In mathematics, the multiple zeta functions are generalizations of the Riemann zeta function, defined by : \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(s 1 ) + ... + 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 s 1 , ..., s k are all positive integers (with s 1 > 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 = s 1 + ... + s k is known as the "weight". 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, : \zeta(2,1,2,1,3) = \zeta(\{2,1\}^2,3).

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