Multiple zeta function

In mathematics, the multiple zeta functions are generalisations of the Riemann zeta function, defined by In mathematics, the multiple zeta functions are generalisations 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. The k in the above definition is named the 'length' of a MZV, and the n = s1 + ... + sk is known as the 'weight'.