Multiple zeta function

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In mathematics, the multiple zeta functions are generalisations 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. [1] [2]

The k in the above definition is named the "length" of a MZV, and the n = s1 + ... + sk is known as the "weight".[3]

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[edit]

In the particular case of only two parameters we have (with s>1 and n,m integer):[4]

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):[4]

s t approximate value explicit formulae OEIS
2 2 0.811742425283353643637002772406 OEISA197110
3 2 0.228810397603353759768746148942
4 2 0.088483382454368714294327839086
5 2 0.038575124342753255505925464373
6 2 0.017819740416835988
2 3 0.711566197550572432096973806086
3 3 0.213798868224592547099583574508
4 3 0.085159822534833651406806018872
5 3 0.037707672984847544011304782294
2 4 0.674523914033968140491560608257
3 4 0.207505014615732095907807605495
4 4 0.083673113016495361614890436542

Note that if we have irreducibles, i.e. these MZVs cannot be written as function of only.[5]

Three parameters case[edit]

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

Euler reflection formula[edit]

The above MZVs satisfy the Euler reflection formula:


Using the shuffle relations, it is easy to prove that:[5]


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

Symmetric sums in terms of the zeta function[edit]

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)[edit]

For any real , .

Proof. Assume the are all distinct. (There is not 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.[6]

For , the theorem says for . This is the main result of.[7]

Having . To state the analog of Theorem 1 for the , we require one bit of notation. For a partition

or , let .

Theorem 2(Hoffman)[edit]

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 (1)

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 (1) 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 .[6]

The sum and duality conjectures[6][edit]

We first state the sum conjecture, which is due to C. Moen.[8]

Sum conjecture(Hoffman). 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's paper[9] and has been rediscovered several times, in particular by Williams.[10] Finally, C. Moen[8] 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.[6]

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:[3]

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

Euler sum with all possible alternations of sign[edit]

The Euler sum with alternations of sign appears in studies of the non-alternating Euler sum.[5]


with are the generalized harmonic numbers.

As a variant of the Dirichlet eta function we define


Reflection formula[edit]

The reflection formula can be generalized as follows:

if we have

Other relations[edit]

Using the series definition it is easy to prove:


A further useful relation is:[5]

where and

Note that must be used for all value for whom the argument of the factorials is

Other results[edit]

For any integer positive ::

or more generally:

Mordell–Tornheim zeta values[edit]

The Mordell–Tornheim zeta function, introduced by Matsumoto (2003) who was motivated by the papers Mordell (1958) and Tornheim (1950), is defined by

It is a special case of the Shintani zeta function.



  1. ^ Zhao, Jianqiang (2010). "Standard relations of multiple polylogarithm values at roots of unity". Documenta Mathematica. 15: 1–34. 
  2. ^ Zhao, Jianqiang (2016). Multiple Zeta Functions, Multiple Polylogarithms and Their Special Values. World Scientific Publishing. ISBN 978-981-4689-39-7. 
  3. ^ a b Hoffman, Mike. "Multiple Zeta Values". Mike Hoffman's Home Page. U.S. Naval Academy. Retrieved June 8, 2012. 
  4. ^ a b Borwein, David; Borwein, Jonathan; Bradley, David (September 23, 2004). "Parametric Euler Sum Identities" (PDF). CARMA, AMSI Honours Course. The University of Newcastle. Retrieved June 3, 2012. 
  5. ^ a b c d Broadhurst, D. J. (1996). "On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory.". arXiv:hep-th/9604128Freely accessible. 
  6. ^ a b c d Hoffman, Michael (1992). "Multiple Harmonic Series". Pacific Journal of Mathematics. 152: 276–278. MR 1141796. Zbl 0763.11037. doi:10.2140/pjm.1992.152.275. 
  7. ^ Ramachandra Rao, R. Sita; M. V. Subbarao (1984). "Transformation formulae for multiple series". Pacific Journal of Mathematics. 113: 417–479. doi:10.2140/pjm.1984.113.471. 
  8. ^ a b Moen, C. "Sums of Simple Series". Preprint. 
  9. ^ Euler, L. (1775). "Meditationes circa singulare serierum genus". Novi Comm. Acad. Sci. Petropol. 15 (20): 140–186. 
  10. ^ Williams, G. T. (1958). "On the evaluation of some multiple series". Journal of the London Mathematical Society. 33: 368–371. doi:10.1112/jlms/s1-33.3.368. 

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