Multiple zeta function-The Daily Ledger

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.

Multiple zeta function

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

Multiple zeta function

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,

Multiple zeta function

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 integral identity on products gives

Two parameters case

In the particular case of only two parameters we have (with s > 1 and n, m integers):

where are the generalized harmonic numbers.

Multiple zeta functions are known to satisfy what is known as MZV duality, the simplest case of which is the famous identity of Euler:

where Hn are the harmonic numbers.

Special values of double zeta functions, with s > 0 and even, t > 1 and odd, but s+t = 2N+1 (taking if necessary ζ(0) = 0):

Note that if we have irreducibles, i.e. these MZVs cannot be written as function of only.

Three parameters case

In the particular case of only three parameters we have (with a > 1 and n, j, i integers):

Euler reflection formula

The above MZVs satisfy the Euler reflection formula: for

Using the shuffle relations, it is easy to prove that:

for

This function can be seen as a generalization of the reflection formulas.

Symmetric sums in terms of the zeta function

Let , and for a partition of the set , let . Also, given such a and a k-tuple of exponents, define .

The relations between the and are: and

Theorem 1 (Hoffman) For any real , .

Proof. Assume the are all distinct. (There is no loss of generality, since we can take limits.) The left-hand side can be written as . Now thinking on the symmetric

group as acting on k-tuple of positive integers. A given k-tuple has an isotropy group

and an associated partition of : is the set of equivalence classes of the relation given by iff , and . Now the term occurs on the left-hand side of exactly times. It occurs on the right-hand side in those terms corresponding to partitions that are refinements of : letting denote refinement, occurs times. Thus, the conclusion will follow if for any k-tuple and associated partition . To see this, note that counts the permutations having cycle type specified by : since any elements of has a unique cycle type specified by a partition that refines , the result follows.