In mathematics, the multiple zeta functions are generalizations of the Riemann zeta function, defined by

and converge when Re(s1)+...+Re(si)>i for alli. 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 are nth roots of unity are all nonnegative integers, the values of the multiple polylogarithm are called colored multiple zeta values of level, they are called Euler sums or alternating multiple zeta values they are simply called multiple zeta values. Multiple zeta values are often written

and Euler sums are written

, so for example

1. .

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:

1. .

This result is extremely useful due to a well-known result regarding products of iterated integrals, namely that

1. is the symmetric group symbols.

to be the free monoid to be the free-vector space can be equipped with the shuffle product, turning it into an algebra, and define

1. ,

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):

1. 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):

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:

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

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

Symmetric sums in terms of the zeta function

and a k.

Theorem 1 (Hoffman)

For any real.

. Now thinking on the symmetric

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

is the set of equivalence classes of the relation for any k counts the permutations having cycle type, the result follows.

. This is the main result of.

, we require one bit of notation. For a partition

.

Theorem 2 (Hoffman)

.

(1)

are even, and negative if they are odd.

The sum and duality conjectures

Source:

We first state the sum conjecture, which is due to C. Moen.

Sum conjecture (Hoffman). For positive integers k and n, where the sum is extended over k.

Three remarks concerning this conjecture

This was proved by Euler and has been rediscovered several times, in particular by Williams. Finally, C. Moen has proved the same conjecture for k=3 by lengthy but elementary arguments. For the duality conjecture, we first define an involution of finite sequences be the set of strictly increasing.

as self-dual.

.

This sum conjecture is also known as Sum Theorem, and it may be expressed as follows: the Riemann zeta value of an integer n≥2 is equal to the sum of all the valid (i.e. with s1>1) MZVs of the partitions of length k and weight n, with 1≤k≤n−1. In formula:

For example, with length k = 2 and weight n = 7:

Euler sum with all possible alternations of sign

The Euler sum with alternations of sign appears in studies of the non-alternating Euler sum.

Notation

1. are the generalized harmonic numbers.

As a variant of the Dirichlet eta function we define

Reflection formula

can be generalized as follows:

Other relations

Using the series definition it is easy to prove:

A further useful relation is:

Other results

:

1. or more generally:

Mordell–Tornheim zeta values

The Mordell–Tornheim zeta function, introduced by who was motivated by the papers and , is defined by

It is a special case of the Shintani zeta function.

References

Notes

External links