“Multiple zeta function”的意思、由来-开放百科全书

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

and converge when Re(s₁) + ... + Re(s_i) > 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, ..., s_k 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 + ... + s_k 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

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

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

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]

Note that if

we have

irreducibles, i.e. these MZVs cannot be written as function of

only.^[6]

Three parameters case

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

Euler reflection formula

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

Theorem 1(Hoffman)

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

group

as acting on k-tuple

of positive integers. A given k-tuple

has an isotropy group

and an associated partition

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

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]

Having

. To state the analog of Theorem 1 for the

, we require one bit of notation. For a partition

Theorem 2(Hoffman)

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 (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

The sum and duality conjectures^[6]

This was proved by Euler^[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

defined by

= complement of

arranged in increasing order. The our definition of

for

with

We shall say the sequences

and

are dual to each other, and refer to a sequence fixed by

as self-dual.^[6]

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]

Euler sum with all possible alternations of sign

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

Notation

Reflection formula

Other relations

Other results

Mordell–Tornheim zeta values

The Mordell–Tornheim zeta function, introduced by {{harvtxt|Matsumoto|2003}} who was motivated by the papers {{harvtxt|Mordell|1958}} and {{harvtxt|Tornheim|1950}}, is defined by

References

Notes

1. ^{{cite journal|title=Standard relations of multiple polylogarithm values at roots of unity |first1=Jianqiang |last1=Zhao |journal=Documenta Mathematica |year=2010 |volume=15 |pages=1–34}}
2. ^{{cite book|title=Multiple Zeta Functions, Multiple Polylogarithms and Their Special Values |volume=12 |first1=Jianqiang |last1=Zhao |publisher=World Scientific Publishing |date=2016|isbn=978-981-4689-39-7 |doi=10.1142/9634 |series=Series on Number Theory and its Applications }}
3. ^¹{{cite web |url=http://www.usna.edu/Users/math/meh/mult.html |title=Multiple Zeta Values |first1=Mike |last1=Hoffman |work=Mike Hoffman's Home Page |publisher=U.S. Naval Academy |accessdate=June 8, 2012}}
4. ^¹{{cite web |url=http://carma.newcastle.edu.au/MZVs/parasums.pdf |title=Parametric Euler Sum Identities |first1=David |last1=Borwein |first2=Jonathan |last2=Borwein |first3=David |last3=Bradley |date=September 23, 2004 |work=CARMA, AMSI Honours Course |publisher=The University of Newcastle |accessdate=June 3, 2012}}
5. ^¹²³{{Cite arxiv | last1 =Broadhurst | first1 = D. J. | title = On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory. | eprint=hep-th/9604128 | year =1996 }}
6. ^¹²³{{cite journal|last=Hoffman|first=Michael|title=Multiple Harmonic Series|journal=Pacific Journal of Mathematics|year=1992|volume=152|issue=2|pages=276–278|mr=1141796|url=http://projecteuclid.org/euclid.pjm/1102636166|zbl=0763.11037|doi=10.2140/pjm.1992.152.275}}
7. ^{{cite journal|last=Ramachandra Rao|first=R. Sita|author2=M. V. Subbarao|title=Transformation formulae for multiple series|journal=Pacific Journal of Mathematics|year=1984|volume=113|issue=2|pages=417–479|doi=10.2140/pjm.1984.113.471}}
8. ^¹{{cite journal|last=Moen|first=C.|title=Sums of Simple Series|journal=Preprint}}
9. ^{{cite journal|last=Euler|first=L.|title=Meditationes circa singulare serierum genus|journal=Novi Comm. Acad. Sci. Petropol|year=1775|volume=15|issue=20|pages=140–186}}
10. ^{{cite journal|last=Williams|first=G. T.|title=On the evaluation of some multiple series|journal=Journal of the London Mathematical Society|year=1958|volume=33|issue=3|pages=368–371|doi=10.1112/jlms/s1-33.3.368}}