Multiple zeta function
Generalizations of the Riemann zeta function / From Wikipedia, the free encyclopedia
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In mathematics, the multiple zeta functions are generalizations of the Riemann zeta function, defined by
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. 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 + ... + sk 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,