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{{Short description|Generalizations of the Riemann zeta function}}
{{for|a different but related multiple zeta function|Barnes zeta function}}
In [[mathematics]], the '''multiple zeta functions''' are generalizations of the [[Riemann zeta function]], defined by

:<math>\zeta(s_1,\ldots,s_k) = \sum_{n_1 > n_2 > \cdots > n_k > 0}\ \frac{1}{n_1^{s_1} \cdots n_k^{s_k}} = \sum_{n_1 > n_2 > \cdots > n_k > 0}\ \prod_{i=1}^k \frac{1}{n_i^{s_i}},\!</math>

and [[convergent series|converge]] when Re(''s''1) + ... + Re(''s''''i'') > ''i'' for all ''i''. Like the Riemann zeta function, the multiple zeta functions can be [[Analytic continuation|analytically continued]] to be [[meromorphic function]]s (see, for example, Zhao (1999)). When ''s''1, ..., ''s''''k'' are all positive [[integer]]s (with ''s''1 > 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.<ref name="Zhao2010">{{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|doi=10.4171/dm/291 |doi-access=free |arxiv=0707.1459 }}</ref><ref name="Zhao2016">{{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 }}</ref>

The ''k'' in the above definition is named the "depth" of a MZV, and the ''n'' = ''s''1 + ... + ''s''''k'' is known as the "weight".<ref name="Hofmann">{{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}}</ref>

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,

: <math>\zeta(2,1,2,1,3) = \zeta(\{2,1\}^2,3).</math>

== Definition ==
Multiple zeta functions arise as special cases of the multiple polylogarithms

:<math>\mathrm{Li}_{s_1,\ldots,s_d}(\mu_1,\ldots,\mu_d) = \sum\limits_{k_1>\cdots>k_d>0}\frac{\mu_1^{k_1}\cdots\mu_d^{k_d}}{k_1^{s_1}\cdots k_d^{s_d}}</math>

which are generalizations of the [[polylogarithm]] functions. When all of the <math>\mu_i </math> are ''n''th [[roots of unity]] and the <math>s_i</math> are all nonnegative integers, the values of the multiple polylogarithm are called '''colored multiple zeta values of level''' <math>n</math>. In particular, when <math>n=2</math>, they are called '''Euler sums''' or '''alternating multiple zeta values''', and when <math>n=1</math> they are simply called '''multiple zeta values.''' Multiple zeta values are often written

:<math>\zeta(s_1,\ldots,s_d) = \sum\limits_{k_1 > \cdots > k_d > 0} \frac{1}{k_1^{s_1}\cdots k_d^{s_d}}</math>

and Euler sums are written

:<math>\zeta(s_1,\ldots,s_d;\varepsilon_1,\ldots,\varepsilon_d) = \sum\limits_{k_1 > \cdots > k_d > 0} \frac{\varepsilon_1^{k_1}\cdots \varepsilon^{k_d}}{k_1^{s_1}\cdots k_d^{s_d}}</math>

where <math>\varepsilon_i = \pm 1</math>. Sometimes, authors will write a bar over an <math>s_i</math> corresponding to an <math>\varepsilon_i</math> equal to <math>-1</math>, so for example

:<math>\zeta(\overline{a},b) = \zeta(a,b;-1,1) </math>.

== 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 [[integral]]s. This result is often stated with the use of a convention for iterated integrals, wherein

:<math>\int_0^x f_1(t) dt \cdots f_d(t) dt = \int_0^x f_1(t_1)\left(\int_0^{t_1}f_2(t_2)\left(\int_0^{t_2} \cdots \left( \int_0^{t_d} f_d(t_d) dt_d\right)\right)dt_2\right)dt_1 </math>

Using this convention, the result can be stated as follows:<ref name="Zhao2016" />

:<math>\mathrm{Li}_{s_1,\ldots,s_d}(\mu_1,\ldots,\mu_d) = \int_0^1 \left(\frac{dt}{t}\right)^{s_1-1}\frac{dt}{a_1-t}\cdots \left(\frac{dt}{t}\right)^{s_d-1} \frac{dt}{a_d-t}</math> where <math>a_j = \prod\limits_{i=1}^j \mu_i^{-1}</math> for <math>j = 1,2,\ldots,d</math>.

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

:<math>\left(\int_0^x f_1(t)dt \cdots f_n(t) dt \right)\!\left(\int_0^x f_{n+1}(t)dt\cdots f_m(t) dt \right) =
\sum\limits_{\sigma \in \mathfrak{Sh}_{n,m}}\int_0^x f_{\sigma(1)}(t)\cdots f_{\sigma(m)}(t) </math> where <math>\mathfrak{Sh}_{n,m}=\{\sigma \in S_m \mid \sigma(1)<\cdots<\sigma(n), \sigma(n+1)<\cdots<\sigma(m)\}</math> and <math>S_m</math> is the [[symmetric group]] on <math>m</math> symbols.

To utilize this in the context of multiple zeta values, define <math>X = \{a,b\}</math>, <math>X^*</math> to be the [[free monoid]] generated by <math>X</math> and <math>\mathfrak{A}</math> to be the [[free module|free]] <math>\Q</math>-[[vector space]] generated by <math>X^*</math>. <math>\mathfrak{A}</math> can be equipped with the [[Shuffle algebra|shuffle product]], turning it into an [[algebra over a field|algebra]]. Then, the multiple zeta function can be viewed as an evaluation map, where we identify <math>a = \frac{dt}{t}</math>, <math>b = \frac{dt}{1-t}</math>, and define

:<math>\zeta(\mathbf{w}) = \int_0^1 \mathbf{w}</math> for any <math>\mathbf{w} \in X^*</math>,

which, by the aforementioned integral [[identity (mathematics)|identity]], makes

:<math>\zeta(a^{s_1-1}b\cdots a^{s_d-1}b) = \zeta(s_1,\ldots,s_d).</math>

Then, the integral identity on products gives<ref name="Zhao2016" />

:<math>\zeta(w)\zeta(v) = \zeta(w \text{ ⧢ } v).</math>

==Two parameters case==

In the particular case of only two parameters we have (with ''s'' > 1 and ''n'', ''m'' integers):<ref name="Carca6">{{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}}</ref>

:<math>\zeta(s,t) = \sum_{n > m \geq 1} \ \frac{1}{n^{s} m^{t}} = \sum_{n=2}^{\infty} \frac{1}{n^{s}} \sum_{m=1}^{n-1} \frac{1}{m^t} = \sum_{n=1}^{\infty} \frac{1}{(n+1)^{s}} \sum_{m=1}^{n} \frac{1}{m^t}</math>

:<math>\zeta(s,t) = \sum_{n=1}^\infty \frac{H_{n,t}}{(n+1)^s}</math> where <math>H_{n,t}</math> are the [[harmonic number#Generalized harmonic numbers|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 [[Leonhard Euler|Euler]]:

:<math>\sum_{n=1}^\infty \frac{H_n}{(n+1)^2} = \zeta(2,1) = \zeta(3) = \sum_{n=1}^\infty \frac{1}{n^3},\!</math>

where ''H''''n'' are the [[harmonic number]]s.

Special values of double zeta functions, with ''s'' > 0 and [[parity (mathematics)|even]], ''t'' > 1 and [[parity (mathematics)|odd]], but ''s''+''t'' = 2''N''+1 (taking if necessary ''ζ''(0) = 0):<ref name="Carca6"/>

:<math>\zeta(s,t) = \zeta(s)\zeta(t) + \tfrac{1}{2}\Big[\tbinom{s+t}{s}-1\Big]\zeta(s+t) - \sum_{r=1}^{N-1}\Big[\tbinom{2r}{s-1}+\tbinom{2r}{t-1}\Big]\zeta(2r+1)\zeta(s+t-1-2r)</math>

{| class="wikitable sortable"
! ''s''!!''t''!!approximate value!!explicit formulae!![[OEIS]]
|-
| 2 || 2 || 0.811742425283353643637002772406 || <math>\tfrac{3}{4}\zeta(4)</math> || {{OEIS link|A197110}}
|-
| 3 || 2 || 0.228810397603353759768746148942 || <math>3\zeta(2)\zeta(3)-\tfrac{11}{2}\zeta(5)</math> || {{OEIS link|A258983}}
|-
| 4 || 2 || 0.088483382454368714294327839086 || <math>\left (\zeta(3)\right )^2-\tfrac{4}{3}\zeta(6)</math> || {{OEIS link|A258984}}
|-
| 5 || 2 || 0.038575124342753255505925464373 || <math>5\zeta(2)\zeta(5)+2\zeta(3)\zeta(4)-11\zeta(7)</math> || {{OEIS link|A258985}}
|-
| 6 || 2 || 0.017819740416835988362659530248 || || {{OEIS link|A258947}}
|-
| 2 || 3 || 0.711566197550572432096973806086 || <math>\tfrac{9}{2}\zeta(5)-2\zeta(2)\zeta(3)</math> || {{OEIS link|A258986}}
|-
| 3 || 3 || 0.213798868224592547099583574508 || <math>\tfrac{1}{2}\left (\left (\zeta(3)\right )^2 -\zeta(6)\right )</math> || {{OEIS link|A258987}}
|-
| 4 || 3 || 0.085159822534833651406806018872 || <math>17\zeta(7)-10\zeta(2)\zeta(5)</math> || {{OEIS link|A258988}}
|-
| 5 || 3 || 0.037707672984847544011304782294 || <math>5\zeta(3)\zeta(5)-\tfrac{147}{24}\zeta(8)-\tfrac{5}{2}\zeta(6,2)</math> || {{OEIS link|A258982}}
|-
| 2 || 4 || 0.674523914033968140491560608257 || <math>\tfrac{25}{12}\zeta(6)-\left (\zeta(3)\right )^2</math> || {{OEIS link|A258989}}
|-
| 3 || 4 || 0.207505014615732095907807605495 || <math>10\zeta(2)\zeta(5)+\zeta(3)\zeta(4)-18\zeta(7)</math> || {{OEIS link|A258990}}
|-
| 4 || 4 || 0.083673113016495361614890436542 || <math>\tfrac{1}{2}\left (\left (\zeta(4)\right )^2 -\zeta(8)\right )</math> || {{OEIS link|A258991}}
|}
Note that if <math>s+t=2p+2</math> we have <math>p/3</math> irreducibles, i.e. these MZVs cannot be written as function of <math>\zeta(a)</math> only.<ref name="Broadhurst"/>

==Three parameters case==

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

:<math>\zeta(a,b,c) = \sum_{n > j > i \geq 1}\ \frac{1}{n^{a} j^{b} i^{c}} = \sum_{n=1}^{\infty} \frac{1}{(n+2)^{a}} \sum_{j=1}^n \frac{1}{(j+1)^b} \sum_{i=1}^{j} \frac{1}{(i)^c} = \sum_{n=1}^{\infty} \frac{1}{(n+2)^{a}} \sum_{j=1}^n \frac{H_{j,c}}{(j+1)^b}</math>

==Euler reflection formula==
The above MZVs satisfy the Euler reflection formula:
:<math>\zeta(a,b)+\zeta(b,a)=\zeta(a)\zeta(b)-\zeta(a+b)</math> for <math>a,b>1</math>

Using the shuffle relations, it is easy to [[mathematical proof|prove]] that:<ref name="Broadhurst">{{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 }}</ref>

:<math>\zeta(a,b,c)+\zeta(a,c,b)+\zeta(b,a,c)+\zeta(b,c,a)+\zeta(c,a,b)+\zeta(c,b,a)=\zeta(a)\zeta(b)\zeta(c)+2\zeta(a+b+c)-\zeta(a)\zeta(b+c)-\zeta(b)\zeta(a+c)-\zeta(c)\zeta(a+b)</math> for <math>a,b,c>1</math>

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

==Symmetric sums in terms of the zeta function==

Let <math>S(i_1,i_2,\cdots,i_k) = \sum_{n_1 \geq n_2 \geq\cdots n_k \geq 1}\frac{1}{n_1^{i_1} n_2^{i_2}\cdots n_k^{i_k}}</math>, and for a partition <math>\Pi=\{P_1, P_2, \dots,P_l\}</math> of the set <math>\{1,2,\dots,k\}</math>, let <math>c(\Pi) = (\left|P_1\right|-1)!(\left|P_2\right|-1)!\cdots(\left|P_l\right|-1)!</math>. Also, given such a <math>\Pi</math> and a ''k''-tuple <math>i=\{i_1,...,i_k\}</math> of exponents, define <math>\prod_{s=1}^l \zeta(\sum_{j \in P_s} i_j)</math>.

The relations between the <math>\zeta</math> and <math>S</math> are:
<math> S(i_1,i_2)=\zeta(i_1,i_2)+\zeta(i_1+i_2)</math> and <math> S(i_1,i_2,i_3)=\zeta(i_1,i_2,i_3)+\zeta(i_1+i_2,i_3)+\zeta(i_1,i_2+i_3)+\zeta(i_1+i_2+i_3).</math>

===Theorem 1 (Hoffman)===
For any [[real number|real]] <math>i_1,\cdots,i_k >1,</math>, <math>\sum_{{\sigma \in \Sigma_k}}S(i_{\sigma(1)}, \dots, i_{\sigma(k)}) = \sum_{\text{partitions } \Pi \text{ of } \{1,\dots,k\}}c(\Pi)\zeta(i,\Pi)</math>.

Proof. Assume the <math>i_j</math> are all distinct. (There is no loss of generality, since we can take limits.) The left-hand side can be written as
<math>\sum_{\sigma}\sum_{n_1\geq n_2 \geq \cdots \geq n_k \geq1} \frac{1}{{n^{i_1}}_{\sigma(1)}{n^{i_2}}_{\sigma(2)} \cdots {n^{i_k}}_{\sigma(k)} }</math>. Now thinking on the symmetric

group <math>\Sigma_k</math> as acting on ''k''-tuple <math>n = (1,\cdots,k)</math> of positive integers. A given ''k''-tuple <math>n=(n_1,\cdots,n_k)</math> has an isotropy group

<math>\Sigma_k(n)</math> and an associated partition <math>\Lambda</math> of <math>(1,2,\cdots,k)</math>: <math>\Lambda</math> is the set of [[equivalence class]]es of the [[equivalence relation|relation]]
given by <math>i \sim j</math> iff <math>n_i=n_j</math>, and <math>\Sigma_k(n) = \{\sigma \in \Sigma_k : \sigma(i) \sim \forall i\}</math>. Now the term <math>\frac{1}
{{n^{i_1}}_{\sigma(1)}{n^{i_2}}_{\sigma(2)} \cdots {n^{i_k}}_{\sigma(k)}}</math> occurs on the left-hand side of <math>\sum_{{\sigma \in \Sigma_k}}S(i_{\sigma(1)}, \dots, i_{\sigma(k)}) = \sum_{\text{partitions } \Pi \text{ of } \{1,\dots,k\}}c(\Pi)\zeta(i,\Pi)</math> exactly <math>\left| \Sigma_k(n) \right|</math> times. It occurs on the right-hand side in those terms corresponding to partitions <math>\Pi</math> that are refinements of <math>\Lambda</math>: letting <math>\succeq</math> denote refinement, <math>\frac{1}
{{n^{i_1}}_{\sigma(1)}{n^{i_2}}_{\sigma(2)} \cdots {n^{i_k}}_{\sigma(k)}}</math> occurs <math>\sum_{\Pi\succeq\Lambda}(\Pi)</math> times. Thus, the conclusion will follow if
<math>\left| \Sigma_k(n) \right| =\sum_{\Pi\succeq\Lambda}c(\Pi)</math> for any ''k''-tuple <math>n=\{n_1,\cdots,n_k\}</math> and associated partition <math>\Lambda</math>.
To see this, note that <math>c(\Pi)</math> counts the permutations having [[cycle type]] specified by <math>\Pi</math>: since any elements of <math>\Sigma_k(n)</math> has a unique cycle type specified by a partition that refines <math>\Lambda</math>, the result follows.<ref name=hof>{{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|doi-access=free}}</ref>

For <math>k=3</math>, the theorem says <math>\sum_{{\sigma \in \Sigma_3}}S(i_{\sigma(1)},i_{\sigma(2)},i_{\sigma(3)}) = \zeta(i_1)\zeta(i_2)\zeta(i_3)+\zeta(i_1+i_2)\zeta(i_3)+\zeta(i_1)\zeta(i_2+i_3)+\zeta(i_1+i_3)\zeta(i_2)+2\zeta(i_1+i_2+i_3)</math>
for <math>i_1,i_2,i_3>1</math>. This is the main result of.<ref>{{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|doi-access=free}}</ref>

Having <math>\zeta(i_1,i_2,\cdots,i_k)=\sum_{n_1> n_2>\cdots n_k\geq1}\frac{1}{n_1^{i_1} n_2^{i_2}\cdots n_k^{i_k}}</math>. To state the analog of Theorem 1 for the <math>\zeta's</math>, we require one bit of notation. For a partition

<math>\Pi = \{P_1,\cdots,P_l\}</math> of <math>\{1,2\cdots,k\}</math>, let <math>\tilde{c}(\Pi)=(-1)^{k-l}c(\Pi)</math>.

===Theorem 2 (Hoffman)===
For any real <math>i_1,\cdots,i_k>1</math>, <math>\sum_{{\sigma \in \Sigma_k}}\zeta(i_{\sigma(1)}, \dots, i_{\sigma(k)})=\sum_{\text{partitions } \Pi \text{ of } \{1,\dots,k\}}\tilde{c}(\Pi)\zeta(i,\Pi)</math>.

Proof. We follow the same line of argument as in the preceding proof. The left-hand side is now
<math>\sum_{\sigma}\sum_{n_1 > n_2 > \cdots > n_k \geq1} \frac{1}{{n^{i_1}}_{\sigma(1)}{n^{i_2}}_{\sigma(2)} \cdots {n^{i_k}}_{\sigma(k)} }</math>, and a term <math>\frac{1}{n^{i_1}_{1}n^{i_2}_{2} \cdots n^{i_k}_{k}}</math> occurs on the left-hand since once if all the <math>n_i</math> are distinct, and not at all otherwise. Thus, it suffices to show
<math>\sum_{\Pi\succeq\Lambda}\tilde{c}(\Pi)=\begin{cases} 1,\text{ if } \left| \Lambda \right|=k \\ 0, \text{ otherwise }. \end{cases}</math> (1)

To prove this, note first that the sign of <math>\tilde{c}(\Pi)</math> is positive if the permutations of cycle type <math>\Pi</math> are [[parity of a permutation|even]], and negative if they are [[parity of a permutation|odd]]: thus, the left-hand side of (1) is the signed sum of the number of even and odd permutations in the isotropy group <math>\Sigma_k(n)</math>. But such an isotropy group has equal numbers of even and odd permutations unless it is trivial, i.e. unless the associated partition <math>\Lambda</math> is
<math>\{\{1\},\{2\},\cdots,\{k\}\}</math>.<ref name="hof"/>

==The sum and duality conjectures==
Source:<ref name="hof"/>

We first state the sum conjecture, which is due to C. Moen.<ref name="Moen">{{cite journal|last=Moen|first=C.|title=Sums of Simple Series|journal=Preprint}}</ref>

Sum conjecture (Hoffman). For positive integers ''k'' and ''n'',
<math>\sum_{i_1+\cdots+i_k=n, i_1>1}\zeta(i_1,\cdots,i_k) = \zeta(n)</math>, where the sum is extended over ''k''-tuples <math>i_1,\cdots,i_k</math> of positive integers with <math>i_1>1</math>.

Three remarks concerning this [[conjecture]] are in order. First, it implies
<math>\sum_{i_1+\cdots+i_k=n, i_1>1}S(i_1,\cdots,i_k)={n-1\choose k-1}\zeta(n)</math>. Second, in the case <math>k=2</math> it says that <math>\zeta(n-1,1)+\zeta(n-2,2)+\cdots+\zeta(2,n-2)=\zeta(n)</math>, or using the relation between the <math>\zeta's</math> and <math>S's</math> and Theorem 1, <math>2S(n-1,1)=(n+1)\zeta(n)-\sum_{k=2}^{n-2}\zeta(k)\zeta(n-k).</math>

This was proved by Euler<ref>{{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}}</ref> and has been rediscovered several times, in particular by Williams.<ref>{{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}}</ref> Finally, C. Moen<ref name="Moen"/> has proved the same conjecture for ''k''=3 by lengthy but elementary arguments.
For the duality conjecture, we first define an [[involution (mathematics)|involution]] <math>\tau</math> on the set <math>\Im</math> of finite [[sequence]]s of positive integers whose first element is greater than 1. Let <math>\Tau</math> be the set of [[Sequence#Increasing and decreasing|strictly increasing]] finite sequences of positive integers, and let <math>\Sigma : \Im \rightarrow \Tau</math> be the function that sends a sequence in <math>\Im</math> to its sequence of partial sums. If <math>\Tau_n</math> is the set of sequences in <math>\Tau</math> whose last element is at most <math>n</math>, we have two commuting involutions <math>R_n</math> and <math>C_n</math> on <math>\Tau_n</math> defined by
<math>R_n(a_1,a_2,\dots,a_l)=(n+1-a_l,n+1-a_{l-1},\dots,n+1-a_1)</math> and
<math>C_n(a_1,\dots,a_l)</math> = complement of <math>\{a_1,\dots,a_l\}</math> in <math>\{1,2,\dots,n\}</math> arranged in increasing order. The our definition of <math>\tau</math> is <math>\tau(I) = \Sigma^{-1}R_nC_n\Sigma(I) = \Sigma^{-1}C_nR_n\Sigma(I)</math> for <math>I=(i_1,i_2,\dots,i_k) \in \Im</math> with <math>i_1+\cdots+i_k=n</math>.

For example,
<math>\tau(3,4,1) = \Sigma^{-1}C_8R_8(3,7,8) = \Sigma^{-1}(3,4,5,7,8) = (3,1,1,2,1).</math>
We shall say the sequences <math>(i_1,\dots,i_k)</math> and <math>\tau(i_1,\dots,i_k)</math> are dual to each other, and refer to a sequence fixed by <math>\tau</math> as self-dual.<ref name="hof"/>

Duality conjecture (Hoffman). If <math>(h_1,\dots,h_{n-k})</math> is dual to <math>(i_1,\dots,i_k)</math>, then <math>\zeta(h_1,\dots,h_{n-k}) = \zeta(i_1,\dots,i_k)</math>.

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 ''s''1 > 1) MZVs of the [[Partition (number theory)|partitions]] of length ''k'' and weight ''n'', with 1 ≤ ''k'' ≤ ''n'' − 1. In formula:<ref name="Hofmann"/>

:<math>\sum_\stackrel{s_1 + \cdots + s_k=n}{s_1>1}\zeta(s_1, \ldots, s_k) = \zeta(n).</math>

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

:<math>\zeta(6,1)+\zeta(5,2)+\zeta(4,3)+\zeta(3,4)+\zeta(2,5) = \zeta(7).</math>

==Euler sum with all possible alternations of sign==
The Euler sum with alternations of sign appears in studies of the non-alternating Euler sum.<ref name="Broadhurst"/>

===Notation===
:<math>\sum_{n=1}^\infty \frac{H_n^{(b)}(-1)^{(n+1)}}{(n+1)^a}=\zeta(\bar{a},b) </math> with <math> H_n^{(b)}=+1+\frac{1}{2^b}+\frac{1}{3^b}+\cdots</math> are the [[harmonic number#Generalized harmonic numbers|generalized harmonic numbers]].
:<math>\sum_{n=1}^\infty \frac{\bar{H}_n^{(b)}}{(n+1)^a}=\zeta(a,\bar{b}) </math> with <math> \bar{H}_n^{(b)}=-1+\frac{1}{2^b}-\frac{1}{3^b}+\cdots</math>
:<math>\sum_{n=1}^\infty \frac{\bar{H}_n^{(b)}(-1)^{(n+1)}}{(n+1)^a}=\zeta(\bar{a},\bar{b}) </math>
:<math>\sum_{n=1}^\infty \frac{(-1)^{n}}{(n+2)^a}\sum_{n=1}^\infty \frac{\bar{H}_n^{(c)}(-1)^{(n+1)}}{(n+1)^b} =\zeta(\bar{a},\bar{b},\bar{c})</math> with <math> \bar{H}_n^{(c)}=-1+\frac{1}{2^c}-\frac{1}{3^c}+\cdots</math>
:<math>\sum_{n=1}^\infty \frac{(-1)^{n}}{(n+2)^a}\sum_{n=1}^\infty \frac{H_n^{(c)}}{(n+1)^b}=\zeta(\bar{a},b,c) </math> with <math> H_n^{(c)}=+1+\frac{1}{2^c}+\frac{1}{3^c}+\cdots</math>
:<math>\sum_{n=1}^\infty \frac{1}{(n+2)^a}\sum_{n=1}^\infty \frac{H_n^{(c)}(-1)^{(n+1)}}{(n+1)^b}=\zeta(a,\bar{b},c) </math>
:<math>\sum_{n=1}^\infty \frac{1}{(n+2)^a}\sum_{n=1}^\infty \frac{\bar{H}_n^{(c)}}{(n+1)^b}=\zeta(a,b,\bar{c}) </math>
As a variant of the [[Dirichlet eta function]] we define
:<math>\phi(s) = \frac{1-2^{(s-1)}} {2^{(s-1)}} \zeta(s)</math> with <math>s>1</math>
:<math>\phi(1) = -\ln 2</math>

===Reflection formula===
The reflection formula <math>\zeta(a,b)+\zeta(b,a)=\zeta(a)\zeta(b)-\zeta(a+b)</math> can be generalized as follows:
:<math>\zeta(a,\bar{b})+\zeta(\bar{b},a)=\zeta(a)\phi(b)-\phi(a+b)</math>
:<math>\zeta(\bar{a},b)+\zeta(b,\bar{a})=\zeta(b)\phi(a)-\phi(a+b)</math>
:<math>\zeta(\bar{a},\bar{b})+\zeta(\bar{b},\bar{a})=\phi(a)\phi(b)-\zeta(a+b)</math>
if <math>a=b</math> we have <math>\zeta(\bar{a},\bar{a})=\tfrac{1}{2}\Big[\phi^2(a)-\zeta(2a)\Big]</math>

===Other relations===
Using the series definition it is easy to prove:
:<math>\zeta(a,b)+\zeta(a,\bar{b})+\zeta(\bar{a},b)+\zeta(\bar{a},\bar{b})=\frac{\zeta(a,b)}{2^{(a+b-2)}}</math> with <math>a>1</math>
:<math>\zeta(a,b,c)+\zeta(a,b,\bar{c})+\zeta(a,\bar{b},c)+\zeta(\bar{a},b,c)+\zeta(a,\bar{b},\bar{c})+\zeta(\bar{a},b,\bar{c})+\zeta(\bar{a},\bar{b},c)+\zeta(\bar{a},\bar{b},\bar{c})=\frac{\zeta(a,b,c)}{2^{(a+b+c-3)}}</math> with <math>a>1</math>
A further useful relation is:<ref name="Broadhurst"/>
:<math>\zeta(a,b)+\zeta(\bar{a},\bar{b})=\sum_{s>0} (a+b-s-1)!\Big[\frac{Z_a(a+b-s,s)}{(a-s)!(b-1)!}+\frac{Z_b(a+b-s,s)}{(b-s)!(a-1)!}\Big]</math>
where <math>Z_a(s,t)=\zeta(s,t)+\zeta(\bar{s},t)-\frac{\Big[\zeta(s,t)+\zeta(s+t)\Big]}{2^{(s-1)}}</math> and <math>Z_b(s,t)=\frac{\zeta(s,t)}{2^{(s-1)}}</math>

Note that <math>s</math> must be used for all value <math>>1</math> for which the argument of the factorials is <math>\geqslant0</math>

==Other results==

For all positive integers <math>a,b,\dots,k</math>:

:<math>\sum_{n=2}^{\infty} \zeta(n,k) = \zeta(k+1)</math> or more generally:
:<math>\sum_{n=2}^{\infty} \zeta(n,a,b,\dots,k) = \zeta(a+1,b,\dots,k)</math>

:<math>\sum_{n=2}^{\infty} \zeta(n,\bar{k}) = -\phi(k+1)</math>
:<math>\sum_{n=2}^{\infty} \zeta(n,\bar{a},b) = \zeta(\overline{a+1},b)</math>
:<math>\sum_{n=2}^{\infty} \zeta(n,a,\bar{b}) = \zeta(a+1,\bar{b})</math>
:<math>\sum_{n=2}^{\infty} \zeta(n,\bar{a},\bar{b}) = \zeta(\overline{a+1},\bar{b})</math>

:<math>\lim_{k \to \infty}\zeta(n,k) = \zeta(n)-1</math>

:<math>1-\zeta(2)+\zeta(3)-\zeta(4)+\cdots=|\frac{1}{2}|</math>

:<math>\zeta(a,a)=\tfrac{1}{2}\Big[(\zeta(a))^{2}-\zeta(2a)\Big]</math>

:<math>\zeta(a,a,a)=\tfrac{1}{6}(\zeta(a))^{3}+\tfrac{1}{3}\zeta(3a)-\tfrac{1}{2}\zeta(a)\zeta(2a)</math>

==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
:<math>\zeta_{MT,r}(s_1,\dots,s_r;s_{r+1})=\sum_{m_1,\dots,m_r>0}\frac{1}{ m_1^{s_1}\cdots m_r^{s_r}(m_1+\dots+m_r)^{s_{r+1}}}</math>
It is a special case of the [[Shintani zeta function]].

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==Notes==
{{Reflist}}

==External links==
* {{cite web
|first1=Jonathan
|last1=Borwein
|first2=Wadim
|last2=Zudilin
|url=http://carma.newcastle.edu.au/MZVs/
|title=Lecture notes on the Multiple Zeta Function}}
*{{cite web
|first1=Michael
|last1=Hoffman
|url=http://www.usna.edu/Users/math/meh/mult.html
|title=Multiple zeta values
|year=2012
}}
*{{cite book
|first1=Jianqiang
|last1=Zhao
|date=2016
|title=Multiple Zeta Functions, Multiple Polylogarithms and Their Special Values
|volume=12
|publisher=World Scientific Publishing
|isbn=978-981-4689-39-7 |doi=10.1142/9634
|series=Series on Number Theory and its Applications
}}
*{{cite web
|first1=José Ignacio
|last1=Burgos Gil
|first2=Javier
|last2=Fresán
|title=Multiple zeta values: from numbers to motives
|url=http://javier.fresan.perso.math.cnrs.fr/mzv.pdf}}

[[Category:Zeta and L-functions]]

Multiple zeta function - Revision history

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