Multiple zeta function


In mathematics, the multiple zeta functions are generalisations of the Riemann zeta function, defined by
and converge when Re + ... + Re > i for all i. Like the Riemann zeta function, the multiple zeta functions can be analytically continued to be meromorphic functions. When s1,..., sk are all positive integers these sums are often called multiple zeta values 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".
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,

Two parameters case

In the particular case of only two parameters we have :
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 :
stapproximate valueexplicit formulaeOEIS
220.811742425283353643637002772406
320.228810397603353759768746148942
420.088483382454368714294327839086
520.038575124342753255505925464373
620.017819740416835988
230.711566197550572432096973806086
330.213798868224592547099583574508A258987
430.085159822534833651406806018872A258988
530.037707672984847544011304782294A258982
240.674523914033968140491560608257A258989
340.207505014615732095907807605495A258990
440.083673113016495361614890436542A258991

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 :

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 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. 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.
For, the theorem says
for. This is the main result of.
Having. To state the analog of Theorem 1 for the, we require one bit of notation. For a partition
or, 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 are distinct, and not at all otherwise. Thus, it suffices to show

To prove this, note first that the sign of is positive if the permutations of cycle-type are even, and negative if they are odd: thus, the left-hand side of is the signed sum of the number of even and odd permutations in the isotropy group. But such an isotropy group has equal numbers of even and odd permutations unless it is trivial, i.e. unless the associated partition is

The sum and duality conjectures

We first state the sum conjecture, which is due to C. Moen.
Sum conjecture. For positive integers k and n,
, where the sum is extended over k-tuples of positive integers with.
Three remarks concerning this conjecture are in order. First, it implies
. Second, in the case it says that, or using the relation between the and 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 on the set of finite sequences of positive integers whose first element is greater than 1. Let be the set of strictly increasing finite sequences of positive integers, and let be the function that sends a sequence in to its sequence of partial sums. If is the set of sequences in whose last element is at most, we have two commuting involutions and on defined by
and
= complement of in arranged in increasing order. The our definition of is for with.
For example,
We shall say the sequences and are dual to each other, and refer to a sequence fixed by as self-dual.
Duality conjecture. 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 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 we have

Other relations

Using the series definition it is easy to prove:
A further useful relation is:
where and
Note that must be used for all value for whom the argument of the factorials is

Other results

For any integer positive ::

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.