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 \mu_i are nth roots of unity and the s_i are all nonnegative integers, the values of the multiple polylogarithm are called colored multiple zeta values of level n. In particular, when n=2, they are called Euler sums or alternating multiple zeta values, and when n=1 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 s_i corresponding to an equal to -1, 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: This result is extremely useful due to a well-known result regarding products of iterated integrals, namely that To utilize this in the context of multiple zeta values, define X = {a,b}, X^* to be the free monoid generated by X and to be the free \Q-vector space generated by X^*. 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 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): 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 s+t=2p+2 we have p/3 irreducibles, i.e. these MZVs cannot be written as function of \zeta(a) 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

Let, and for a partition of the set , let. Also, given such a \Pi and a k-tuple of exponents, define. The relations between the \zeta and S are: and

Theorem 1 (Hoffman)

For any real,. Proof. Assume the i_j 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 \Sigma_k as acting on k-tuple of positive integers. A given k-tuple has an isotropy group \Sigma_k(n) and an associated partition \Lambda of : \Lambda is the set of equivalence classes of the relation given by i \sim j iff n_i=n_j, 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 \Pi that are refinements of \Lambda: letting \succeq denote refinement, occurs times. Thus, the conclusion will follow if for any k-tuple and associated partition \Lambda. To see this, note that c(\Pi) counts the permutations having cycle type specified by \Pi: since any elements of \Sigma_k(n) has a unique cycle type specified by a partition that refines \Lambda, the result follows. For k=3, the theorem says for. This is the main result of. Having. To state the analog of Theorem 1 for the \zeta's, we require one bit of notation. For a partition of, let.

Theorem 2 (Hoffman)

For any real,. Proof. We follow the same line of argument as in the preceding proof. The left-hand side is now , and a term occurs on the left-hand since once if all the n_i are distinct, and not at all otherwise. Thus, it suffices to show (1) To prove this, note first that the sign of is positive if the permutations of cycle type \Pi are even, and negative if they are odd: thus, the left-hand side of (1) is the signed sum of the number of even and odd permutations in the isotropy group \Sigma_k(n). But such an isotropy group has equal numbers of even and odd permutations unless it is trivial, i.e. unless the associated partition \Lambda is.

The sum and duality conjectures

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-tuples of positive integers with i_1>1. Three remarks concerning this conjecture are in order. First, it implies . Second, in the case k=2 it says that, or using the relation between the \zeta's and S's and Theorem 1, 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 \tau on the set \Im of finite sequences of positive integers whose first element is greater than 1. Let \Tau be the set of strictly increasing finite sequences of positive integers, and let be the function that sends a sequence in \Im to its sequence of partial sums. If \Tau_n is the set of sequences in \Tau whose last element is at most n, we have two commuting involutions R_n and C_n on \Tau_n defined by and = complement of in arranged in increasing order. The our definition of \tau is for with. For example, We shall say the sequences and are dual to each other, and refer to a sequence fixed by \tau as self-dual. Duality conjecture (Hoffman). If is dual to, then. 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

As a variant of the Dirichlet eta function we define

Reflection formula

The reflection formula can be generalized as follows: if a=b we have

Other relations

Using the series definition it is easy to prove: A further useful relation is: where and Note that s must be used for all value >1 for which the argument of the factorials is \geqslant0

Other results

For all positive integers a,b,\dots,k:

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.