Multiple zeta function

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.[1][2]

The k in the above definition is named the "depth" 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,

Definition

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Multiple zeta functions arise as special cases of the multiple polylogarithms

 

which are generalizations of the polylogarithm functions. When all of the   are nth roots of unity and the   are all nonnegative integers, the values of the multiple polylogarithm are called colored multiple zeta values of level  . In particular, when  , they are called Euler sums or alternating multiple zeta values, and when   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   corresponding to an   equal to  , so for example

 .

Integral structure and identities

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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:[2]

  where   for  .

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

  where   and   is the symmetric group on   symbols.

To utilize this in the context of multiple zeta values, define  ,   to be the free monoid generated by   and   to be the free  -vector space generated by  .   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

  for any  ,

which, by the aforementioned integral identity, makes

 

Then, the integral identity on products gives[2]

 

Two parameters case

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In the particular case of only two parameters we have (with s > 1 and n, m integers):[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   A197110
3 2 0.228810397603353759768746148942   A258983
4 2 0.088483382454368714294327839086   A258984
5 2 0.038575124342753255505925464373   A258985
6 2 0.017819740416835988362659530248 A258947
2 3 0.711566197550572432096973806086   A258986
3 3 0.213798868224592547099583574508   A258987
4 3 0.085159822534833651406806018872   A258988
5 3 0.037707672984847544011304782294   A258982
2 4 0.674523914033968140491560608257   A258989
3 4 0.207505014615732095907807605495   A258990
4 4 0.083673113016495361614890436542   A258991

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

Three parameters case

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

 

Euler reflection formula

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The above MZVs satisfy the Euler reflection formula:

  for  

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

  for  

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

Symmetric sums in terms of the zeta function

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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)

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For any real  ,  .

Proof. Assume the   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   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

  of  , let  .

Theorem 2 (Hoffman)

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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]

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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[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

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

Notation

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  with   are the generalized harmonic numbers.
  with  
 
  with  
  with  
 
 

As a variant of the Dirichlet eta function we define

  with  
 

Reflection formula

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The reflection formula   can be generalized as follows:

 
 
 

if   we have  

Other relations

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Using the series definition it is easy to prove:

  with  
  with  

A further useful relation is:[5]

 

where   and  

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

Other results

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For all positive integers  :

  or more generally:
 
 
 
 
 
 
 
 
 

Mordell–Tornheim zeta values

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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.

References

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  • Tornheim, Leonard (1950). "Harmonic double series". American Journal of Mathematics. 72 (2): 303–314. doi:10.2307/2372034. ISSN 0002-9327. JSTOR 2372034. MR 0034860.
  • Mordell, Louis J. (1958). "On the evaluation of some multiple series". Journal of the London Mathematical Society. Second Series. 33 (3): 368–371. doi:10.1112/jlms/s1-33.3.368. ISSN 0024-6107. MR 0100181.
  • Apostol, Tom M.; Vu, Thiennu H. (1984), "Dirichlet series related to the Riemann zeta function", Journal of Number Theory, 19 (1): 85–102, doi:10.1016/0022-314X(84)90094-5, ISSN 0022-314X, MR 0751166
  • Crandall, Richard E.; Buhler, Joe P. (1994). "On the evaluation of Euler Sums". Experimental Mathematics. 3 (4): 275. doi:10.1080/10586458.1994.10504297. MR 1341720.
  • Borwein, Jonathan M.; Girgensohn, Roland (1996). "Evaluation of Triple Euler Sums". Electron. J. Comb. 3 (1): #R23. doi:10.37236/1247. hdl:1959.13/940394. MR 1401442.
  • Flajolet, Philippe; Salvy, Bruno (1998). "Euler Sums and contour integral representations". Exp. Math. 7: 15–35. CiteSeerX 10.1.1.37.652. doi:10.1080/10586458.1998.10504356.
  • Zhao, Jianqiang (1999). "Analytic continuation of multiple zeta functions". Proceedings of the American Mathematical Society. 128 (5): 1275–1283. doi:10.1090/S0002-9939-99-05398-8. MR 1670846.
  • Matsumoto, Kohji (2003), "On Mordell–Tornheim and other multiple zeta-functions", Proceedings of the Session in Analytic Number Theory and Diophantine Equations, Bonner Math. Schriften, vol. 360, Bonn: Univ. Bonn, MR 2075634
  • Espinosa, Olivier; Moll, Victor Hugo (2008). "The evaluation of Tornheim double sums". arXiv:math/0505647.
  • Espinosa, Olivier; Moll, Victor Hugo (2010). "The evaluation of Tornheim double sums II". Ramanujan J. 22: 55–99. arXiv:0811.0557. doi:10.1007/s11139-009-9181-1. MR 2610609. S2CID 17055581.
  • Borwein, J.M.; Chan, O-Y. (2010). "Duality in tails of multiple zeta values". Int. J. Number Theory. 6 (3): 501–514. CiteSeerX 10.1.1.157.9158. doi:10.1142/S1793042110003058. MR 2652893.
  • Basu, Ankur (2011). "On the evaluation of Tornheim sums and allied double sums". Ramanujan J. 26 (2): 193–207. doi:10.1007/s11139-011-9302-5. MR 2853480. S2CID 120229489.

Notes

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  1. ^ Zhao, Jianqiang (2010). "Standard relations of multiple polylogarithm values at roots of unity". Documenta Mathematica. 15: 1–34. arXiv:0707.1459.
  2. ^ a b c Zhao, Jianqiang (2016). Multiple Zeta Functions, Multiple Polylogarithms and Their Special Values. Series on Number Theory and its Applications. Vol. 12. World Scientific Publishing. doi:10.1142/9634. 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/9604128.
  6. ^ a b c d Hoffman, Michael (1992). "Multiple Harmonic Series". Pacific Journal of Mathematics. 152 (2): 276–278. doi:10.2140/pjm.1992.152.275. MR 1141796. Zbl 0763.11037.
  7. ^ Ramachandra Rao, R. Sita; M. V. Subbarao (1984). "Transformation formulae for multiple series". Pacific Journal of Mathematics. 113 (2): 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 (3): 368–371. doi:10.1112/jlms/s1-33.3.368.
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