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. The k in the above definition is named the "depth" 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, Definition Multiple zeta functions arise as special cases of the multiple polylogarithms which are generalizations of the polylogarithm functions. When all of the are nth roots of unity and the are all nonnegative integers, the values of the multiple polylogarithm are called colored multiple zeta values of level . In particular, when , they are called Euler sums or alternating multiple zeta values, and when they are simply called multiple zeta values. Multiple zeta values are often written and Euler sums are written where . Sometimes, authors will write a bar over an corresponding to an equal to , so for example . Integral structure and identities It was noticed by Kontsevich that it is possible to express colored multiple zeta values (and thus their special cases) as certain multivariable integrals. This result is often stated with the use of a convention for iterated integrals, wherein Using this convention, the result can be stated as follows: where for . This result is extremely useful due to a well-known result regarding products of iterated integrals, namely that where and is the symmetric group on symbols. To utilize this in the context of multiple zeta values, define , to be the free monoid generated by and to be the free -vector space generated by . can be equipped with the shuffle product, turning it into an algebra. Then, the multiple zeta function can be viewed as an evaluation map, where we identify , , and define for any , which, by the aforementioned integral identity, makes Then, the