Multiple Zeta Function

Multiple Zeta Function

In mathematics, the multiple zeta functions are generalisations 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(s1) + ... + Re(si) > 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, ..., sk are all positive integers (with s1 > 1) these sums are often called multiple zeta values (MZVs) or Euler sums.

The k in the above definition is named the "lengh" of a MZV, and the n = s1 + ... + sk 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,

Read more about Multiple Zeta Function:  Two Parameters Case, Three Parameters Case, Euler Reflection Formula, Symmetric Sums in Terms of The Zeta Function, The Sum and Duality Conjectures, Other Results, Mordell–Tornheim Zeta Values, References

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