# Multiple zeta function > Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Multiple_zeta_function > Markdown URL: https://mediated.wiki/source/Multiple_zeta_function.md > Source: https://en.wikipedia.org/wiki/Multiple_zeta_function > Source revision: 1345280973 > License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) {{Short description|Generalizations of the Riemann zeta function}} {{for|a different but related multiple zeta function|Barnes zeta function}} In [mathematics](/source/mathematics), the '''multiple zeta functions''' are generalizations of the [Riemann zeta function](/source/Riemann_zeta_function), defined by :

\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}},\!

and [converge](/source/convergent_series) when Re(''s''₁) + ... + Re(''s''_''i'') > ''i'' for all ''i''. Like the Riemann zeta function, the multiple zeta functions can be [analytically continued](/source/Analytic_continuation) to be [meromorphic function](/source/meromorphic_function)s (see, for example, Zhao (1999)). When ''s''₁, ..., ''s''_''k'' are all positive [integer](/source/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.{{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 }}{{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 }} 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".{{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}} 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, :

\zeta(2,1,2,1,3) = \zeta(\{2,1\}^2,3).

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

\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}}

which are generalizations of the [polylogarithm](/source/polylogarithm) functions. When all of the

\mu_i

are ''n''^th [roots of unity](/source/roots_of_unity) and the

s_i

are all nonnegative integers, the values of the multiple polylogarithm are called '''colored multiple zeta values of level'''

n

. In particular, when

n=2

, they are called '''Euler sums''' or '''alternating multiple zeta values''', and when

n=1

they are simply called '''multiple zeta values.''' Multiple zeta values are often written :

\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}}

and Euler sums are written :

\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_d^{k_d}}{k_1^{s_1}\cdots k_d^{s_d}}

where

\varepsilon_i = \pm 1

. Sometimes, authors will write a bar over an

s_i

corresponding to an

\varepsilon_i

equal to

-1

, so for example :

\zeta(\overline{a},b) = \zeta(a,b;-1,1)

. == 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](/source/integral)s. This result is often stated with the use of a convention for iterated integrals, wherein :

\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

Using this convention, the result can be stated as follows: :

\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}

where

a_j = \prod\limits_{i=1}^j \mu_i^{-1}

for

j = 1,2,\ldots,d

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

\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)

where

\mathfrak{Sh}_{n,m}=\{\sigma \in S_m \mid \sigma(1)<\cdots<\sigma(n), \sigma(n+1)<\cdots<\sigma(m)\}

and

S_m

is the [symmetric group](/source/symmetric_group) on

m

symbols. To utilize this in the context of multiple zeta values, define

X = \{a,b\}

X^*

to be the [free monoid](/source/free_monoid) generated by

X

and

\mathfrak{A}

to be the [free](/source/free_module)

\Q

-[vector space](/source/vector_space) generated by

X^*

\mathfrak{A}

can be equipped with the [shuffle product](/source/Shuffle_algebra), turning it into an [algebra](/source/algebra_over_a_field). Then, the multiple zeta function can be viewed as an evaluation map, where we identify

a = \frac{dt}{t}

b = \frac{dt}{1-t}

, and define :

\zeta(\mathbf{w}) = \int_0^1 \mathbf{w}

for any

\mathbf{w} \in X^*

, which, by the aforementioned integral [identity](/source/identity_(mathematics)), makes :

\zeta(a^{s_1-1}b\cdots a^{s_d-1}b) = \zeta(s_1,\ldots,s_d).

Then, the integral identity on products gives :

\zeta(w)\zeta(v) = \zeta(w \text{ ⧢ } v).

==Two parameters case== In the particular case of only two parameters we have (with ''s'' > 1 and ''n'', ''m'' integers):{{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}} :

\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}

\zeta(s,t) = \sum_{n=1}^\infty \frac{H_{n,t}}{(n+1)^s}

where

H_{n,t}

are the [generalized harmonic numbers](/source/harmonic_number). Multiple zeta functions are known to satisfy what is known as MZV duality, the simplest case of which is the famous identity of [Euler](/source/Leonhard_Euler): :

\sum_{n=1}^\infty \frac{H_n}{(n+1)^2} = \zeta(2,1) = \zeta(3) = \sum_{n=1}^\infty \frac{1}{n^3},\!

where ''H''_''n'' are the [harmonic number](/source/harmonic_number)s. Special values of double zeta functions, with ''s'' > 0 and [even](/source/parity_(mathematics)), ''t'' > 1 and [odd](/source/parity_(mathematics)), but ''s''+''t'' = 2''N''+1 (taking if necessary ''ζ''(0) = 0): :

\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)

{| class="wikitable sortable" ! ''s''!!''t''!!approximate value!!explicit formulae!![OEIS](/source/OEIS) |- | 2 || 2 || 0.811742425283353643637002772406 ||

\tfrac{3}{4}\zeta(4)

|| {{OEIS link|A197110}} |- | 3 || 2 || 0.228810397603353759768746148942 ||

3\zeta(2)\zeta(3)-\tfrac{11}{2}\zeta(5)

|| {{OEIS link|A258983}} |- | 4 || 2 || 0.088483382454368714294327839086 ||

\left (\zeta(3)\right )^2-\tfrac{4}{3}\zeta(6)

|| {{OEIS link|A258984}} |- | 5 || 2 || 0.038575124342753255505925464373 ||

5\zeta(2)\zeta(5)+2\zeta(3)\zeta(4)-11\zeta(7)

|| {{OEIS link|A258985}} |- | 6 || 2 || 0.017819740416835988362659530248 || || {{OEIS link|A258947}} |- | 2 || 3 || 0.711566197550572432096973806086 ||

\tfrac{9}{2}\zeta(5)-2\zeta(2)\zeta(3)

|| {{OEIS link|A258986}} |- | 3 || 3 || 0.213798868224592547099583574508 ||

\tfrac{1}{2}\left (\left (\zeta(3)\right )^2 -\zeta(6)\right )

|| {{OEIS link|A258987}} |- | 4 || 3 || 0.085159822534833651406806018872 ||

17\zeta(7)-10\zeta(2)\zeta(5)

|| {{OEIS link|A258988}} |- | 5 || 3 || 0.037707672984847544011304782294 ||

5\zeta(3)\zeta(5)-\tfrac{147}{24}\zeta(8)-\tfrac{5}{2}\zeta(6,2)

|| {{OEIS link|A258982}} |- | 2 || 4 || 0.674523914033968140491560608257 ||

\tfrac{25}{12}\zeta(6)-\left (\zeta(3)\right )^2

|| {{OEIS link|A258989}} |- | 3 || 4 || 0.207505014615732095907807605495 ||

10\zeta(2)\zeta(5)+\zeta(3)\zeta(4)-18\zeta(7)

|| {{OEIS link|A258990}} |- | 4 || 4 || 0.083673113016495361614890436542 ||

\tfrac{1}{2}\left (\left (\zeta(4)\right )^2 -\zeta(8)\right )

|| {{OEIS link|A258991}} |} Note that if

s+t=2p+2

we have

p/3

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

\zeta(a)

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

\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}

==Euler reflection formula== The above MZVs satisfy the Euler [reflection formula](/source/reflection_formula): :

\zeta(a,b)+\zeta(b,a)=\zeta(a)\zeta(b)-\zeta(a+b)

for

a,b>1

Using the shuffle relations, it is easy to [prove](/source/mathematical_proof) that:{{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 }} :

\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)

for

a,b,c>1

This function can be seen as a generalization of the reflection formulas. ==Symmetric sums in terms of the zeta function== Let

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}}

, and for a partition

\Pi=\{P_1, P_2, \dots,P_l\}

of the set

\{1,2,\dots,k\}

, let

c(\Pi) = (\left|P_1\right|-1)!(\left|P_2\right|-1)!\cdots(\left|P_l\right|-1)!

. Also, given such a

\Pi

and a ''k''-tuple

i=\{i_1,...,i_k\}

of exponents, define

\prod_{s=1}^l \zeta(\sum_{j \in P_s} i_j)

. The relations between the

\zeta

and

S

are:

S(i_1,i_2)=\zeta(i_1,i_2)+\zeta(i_1+i_2)

and

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).

===Theorem 1 (Hoffman)=== For any [real](/source/real_number)

i_1,\cdots,i_k >1,

\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)

. Proof. Assume the

i_j

are all distinct. (There is no loss of generality, since we can take limits.) The left-hand side can be written as

\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)} }

. Now thinking on the symmetric group

\Sigma_k

as acting on ''k''-tuple

n = (1,\cdots,k)

of positive integers. A given ''k''-tuple

n=(n_1,\cdots,n_k)

has an [isotropy](/source/isotropy) group

\Sigma_k(n)

and an associated partition

\Lambda

(1,2,\cdots,k)

\Lambda

is the set of [equivalence class](/source/equivalence_class)es of the [relation](/source/equivalence_relation) given by

i \sim j

iff

n_i=n_j

, and

\Sigma_k(n) = \{\sigma \in \Sigma_k : \sigma(i) \sim \forall i\}

. Now the term

\frac{1}
{{n^{i_1}}_{\sigma(1)}{n^{i_2}}_{\sigma(2)} \cdots {n^{i_k}}_{\sigma(k)}}

occurs on the left-hand side of

\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)

exactly

\left| \Sigma_k(n) \right|

times. It occurs on the right-hand side in those terms corresponding to partitions

\Pi

that are refinements of

\Lambda

: letting

\succeq

denote refinement,

\frac{1}
{{n^{i_1}}_{\sigma(1)}{n^{i_2}}_{\sigma(2)} \cdots {n^{i_k}}_{\sigma(k)}}

occurs

\sum_{\Pi\succeq\Lambda}(\Pi)

times. Thus, the conclusion will follow if

\left| \Sigma_k(n) \right| =\sum_{\Pi\succeq\Lambda}c(\Pi)

for any ''k''-tuple

n=\{n_1,\cdots,n_k\}

and associated partition

\Lambda

. To see this, note that

c(\Pi)

counts the permutations having [cycle type](/source/cycle_type) specified by

\Pi

: since any elements of

\Sigma_k(n)

has a unique cycle type specified by a partition that refines

\Lambda

, the result follows.{{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}} For

k=3

, the theorem says

\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)

for

i_1,i_2,i_3>1

. This is the main result of.{{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}} Having

\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}}

. To state the analog of Theorem 1 for the

\zeta's

, we require one bit of notation. For a partition

\Pi = \{P_1,\cdots,P_l\}

\{1,2\cdots,k\}

, let

\tilde{c}(\Pi)=(-1)^{k-l}c(\Pi)

. ===Theorem 2 (Hoffman)=== For any real

i_1,\cdots,i_k>1

\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)

. Proof. We follow the same line of argument as in the preceding proof. The left-hand side is now

\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)} }

, and a term

\frac{1}{n^{i_1}_{1}n^{i_2}_{2} \cdots n^{i_k}_{k}}

occurs on the left-hand since once if all the

n_i

are distinct, and not at all otherwise. Thus, it suffices to show

\sum_{\Pi\succeq\Lambda}\tilde{c}(\Pi)=\begin{cases} 1,\text{ if } \left| \Lambda \right|=k \\ 0, \text{ otherwise }. \end{cases}

(1) To prove this, note first that the sign of

\tilde{c}(\Pi)

is positive if the permutations of cycle type

\Pi

are [even](/source/parity_of_a_permutation), and negative if they are [odd](/source/parity_of_a_permutation): thus, the left-hand side of (1) is the signed sum of the number of even and odd permutations in the isotropy group

\Sigma_k(n)

. But such an isotropy group has equal numbers of even and odd permutations unless it is trivial, i.e. unless the associated partition

\Lambda

\{\{1\},\{2\},\cdots,\{k\}\}

. ==The sum and duality conjectures== Source: We first state the sum conjecture, which is due to C. Moen.{{cite journal|last=Moen|first=C.|title=Sums of Simple Series|journal=Preprint}} Sum conjecture (Hoffman). For positive integers ''k'' and ''n'',

\sum_{i_1+\cdots+i_k=n, i_1>1}\zeta(i_1,\cdots,i_k) = \zeta(n)

, where the sum is extended over ''k''-tuples

i_1,\cdots,i_k

of positive integers with

i_1>1

. Three remarks concerning this [conjecture](/source/conjecture) are in order. First, it implies

\sum_{i_1+\cdots+i_k=n, i_1>1}S(i_1,\cdots,i_k)={n-1\choose k-1}\zeta(n)

. Second, in the case

k=2

it says that

\zeta(n-1,1)+\zeta(n-2,2)+\cdots+\zeta(2,n-2)=\zeta(n)

, or using the relation between the

\zeta's

and

S's

and Theorem 1,

2S(n-1,1)=(n+1)\zeta(n)-\sum_{k=2}^{n-2}\zeta(k)\zeta(n-k).

This was proved by Euler{{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}} and has been rediscovered several times, in particular by Williams.{{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}} 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](/source/involution_(mathematics))

\tau

on the set

\Im

of finite [sequence](/source/sequence)s of positive integers whose first element is greater than 1. Let

\Tau

be the set of [strictly increasing](/source/Sequence) finite sequences of positive integers, and let

\Sigma : \Im \rightarrow \Tau

be the function that sends a sequence in

\Im

to its sequence of partial sums. If

\Tau_n

is the set of sequences in

\Tau

whose last element is at most

n

, we have two commuting involutions

R_n

and

C_n

\Tau_n

defined by

R_n(a_1,a_2,\dots,a_l)=(n+1-a_l,n+1-a_{l-1},\dots,n+1-a_1)

and

C_n(a_1,\dots,a_l)

= complement of

\{a_1,\dots,a_l\}

\{1,2,\dots,n\}

arranged in increasing order. The our definition of

\tau

\tau(I) = \Sigma^{-1}R_nC_n\Sigma(I) = \Sigma^{-1}C_nR_n\Sigma(I)

for

I=(i_1,i_2,\dots,i_k) \in \Im

with

i_1+\cdots+i_k=n

. For example,

\tau(3,4,1) = \Sigma^{-1}C_8R_8(3,7,8) = \Sigma^{-1}(3,4,5,7,8) = (3,1,1,2,1).

We shall say the sequences

(i_1,\dots,i_k)

and

\tau(i_1,\dots,i_k)

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

\tau

as self-dual. Duality conjecture (Hoffman). If

(h_1,\dots,h_{n-k})

is dual to

(i_1,\dots,i_k)

, then

\zeta(h_1,\dots,h_{n-k}) = \zeta(i_1,\dots,i_k)

. 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 [partitions](/source/Partition_(number_theory)) of length ''k'' and weight ''n'', with 1 ≤ ''k'' ≤ ''n'' − 1. In formula: :

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

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

\zeta(6,1)+\zeta(5,2)+\zeta(4,3)+\zeta(3,4)+\zeta(2,5) = \zeta(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=== :

\sum_{n=1}^\infty \frac{H_n^{(b)}(-1)^{(n+1)}}{(n+1)^a}=\zeta(\bar{a},b)

with

H_n^{(b)}=+1+\frac{1}{2^b}+\frac{1}{3^b}+\cdots

are the [generalized harmonic numbers](/source/harmonic_number). :

\sum_{n=1}^\infty \frac{\bar{H}_n^{(b)}}{(n+1)^a}=\zeta(a,\bar{b})

with

\bar{H}_n^{(b)}=-1+\frac{1}{2^b}-\frac{1}{3^b}+\cdots

\sum_{n=1}^\infty \frac{\bar{H}_n^{(b)}(-1)^{(n+1)}}{(n+1)^a}=\zeta(\bar{a},\bar{b})

\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})

with

\bar{H}_n^{(c)}=-1+\frac{1}{2^c}-\frac{1}{3^c}+\cdots

\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)

with

H_n^{(c)}=+1+\frac{1}{2^c}+\frac{1}{3^c}+\cdots

\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)

\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})

As a variant of the [Dirichlet eta function](/source/Dirichlet_eta_function) we define :

\phi(s) = \frac{1-2^{(s-1)}} {2^{(s-1)}} \zeta(s)

with

s>1

\phi(1) = -\ln 2

===Reflection formula=== The reflection formula

\zeta(a,b)+\zeta(b,a)=\zeta(a)\zeta(b)-\zeta(a+b)

can be generalized as follows: :

\zeta(a,\bar{b})+\zeta(\bar{b},a)=\zeta(a)\phi(b)-\phi(a+b)

\zeta(\bar{a},b)+\zeta(b,\bar{a})=\zeta(b)\phi(a)-\phi(a+b)

\zeta(\bar{a},\bar{b})+\zeta(\bar{b},\bar{a})=\phi(a)\phi(b)-\zeta(a+b)

a=b

we have

\zeta(\bar{a},\bar{a})=\tfrac{1}{2}\Big[\phi^2(a)-\zeta(2a)\Big]

===Other relations=== Using the series definition it is easy to prove: :

\zeta(a,b)+\zeta(a,\bar{b})+\zeta(\bar{a},b)+\zeta(\bar{a},\bar{b})=\frac{\zeta(a,b)}{2^{(a+b-2)}}

with

a>1

\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)}}

with

a>1

A further useful relation is: :

\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]

where

Z_a(s,t)=\zeta(s,t)+\zeta(\bar{s},t)-\frac{\Big[\zeta(s,t)+\zeta(s+t)\Big]}{2^{(s-1)}}

and

Z_b(s,t)=\frac{\zeta(s,t)}{2^{(s-1)}}

Note that

s

must be used for all value

>1

for which the argument of the factorials is

\geqslant0

==Other results== For all positive integers

a,b,\dots,k

: :

\sum_{n=2}^{\infty} \zeta(n,k) = \zeta(k+1)

or more generally: :

\sum_{n=2}^{\infty} \zeta(n,a,b,\dots,k) = \zeta(a+1,b,\dots,k)

\sum_{n=2}^{\infty} \zeta(n,\bar{k}) = -\phi(k+1)

\sum_{n=2}^{\infty} \zeta(n,\bar{a},b) = \zeta(\overline{a+1},b)

\sum_{n=2}^{\infty} \zeta(n,a,\bar{b}) = \zeta(a+1,\bar{b})

\sum_{n=2}^{\infty} \zeta(n,\bar{a},\bar{b}) = \zeta(\overline{a+1},\bar{b})

\lim_{k \to \infty}\zeta(n,k) = \zeta(n)-1

1-\zeta(2)+\zeta(3)-\zeta(4)+\cdots=|\frac{1}{2}|

\zeta(a,a)=\tfrac{1}{2}\Big[(\zeta(a))^{2}-\zeta(2a)\Big]

\zeta(a,a,a)=\tfrac{1}{6}(\zeta(a))^{3}+\tfrac{1}{3}\zeta(3a)-\tfrac{1}{2}\zeta(a)\zeta(2a)

==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 :

\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}}}

It is a special case of the [Shintani zeta function](/source/Shintani_zeta_function). ==References== *{{Cite journal | last1=Tornheim | first1=Leonard | title=Harmonic double series | doi=10.2307/2372034 | mr=0034860 | year=1950 | journal=[American Journal of Mathematics](/source/American_Journal_of_Mathematics) | issn=0002-9327 | volume=72 | issue=2 | pages=303–314| jstor=2372034 }} *{{Cite journal| last1=Mordell | first1=Louis J. | author1-link=Louis Mordell | title=On the evaluation of some multiple series | doi=10.1112/jlms/s1-33.3.368 | mr=0100181 | year=1958 | journal=Journal of the London Mathematical Society | series = Second Series | issn=0024-6107 | volume=33 | issue=3 | pages=368–371}} *{{Citation | last1=Apostol | first1=Tom M. | author1-link=Tom M. Apostol | last2=Vu | first2=Thiennu H. | title=Dirichlet series related to the Riemann zeta function | doi=10.1016/0022-314X(84)90094-5 | year=1984 | journal=[Journal of Number Theory](/source/Journal_of_Number_Theory) | issn=0022-314X | volume=19 | issue=1 | mr=0751166 | pages=85–102| doi-access=free}} * {{cite journal |first1=Richard E. |last1=Crandall |first2=Joe P. |last2= Buhler |title=On the evaluation of Euler Sums |year=1994 |journal=Experimental Mathematics |volume=3 |issue=4 |url=http://www.emis.de/journals/EM/expmath/volumes/3/3.html |page=275 |mr=1341720 |doi=10.1080/10586458.1994.10504297 |url-access=subscription }} * {{cite journal |first1=Jonathan M. |last1=Borwein |first2=Roland |last2=Girgensohn |title=Evaluation of Triple Euler Sums |year=1996 |journal=Electron. J. Comb. |volume=3 |issue=1 |pages=#R23 |doi=10.37236/1247 |mr=1401442 |url=http://www.combinatorics.org/ojs/index.php/eljc/article/view/v3i1r23 |doi-access=free |hdl=1959.13/940394 |hdl-access=free }} *{{cite journal|first1=Philippe|last1=Flajolet | first2=Bruno|last2=Salvy |title=Euler Sums and contour integral representations |journal=Exp. Math. |year=1998 |volume=7 |pages=15–35 |url=http://www.emis.de/journals/EM/expmath/volumes/7/7.html |doi=10.1080/10586458.1998.10504356 |citeseerx=10.1.1.37.652 }} *{{Cite journal| last=Zhao | first=Jianqiang | title=Analytic continuation of multiple zeta functions | doi= 10.1090/S0002-9939-99-05398-8 | year=1999 | journal=Proceedings of the American Mathematical Society | issue=5 | volume=128 | pages=1275–1283 |mr=1670846| doi-access=free}} *{{Citation | last=Matsumoto|first= Kohji |chapter=On Mordell–Tornheim and other multiple zeta-functions|title= Proceedings of the Session in Analytic Number Theory and Diophantine Equations |series=Bonner Math. Schriften|volume= 360|publisher= Univ. Bonn |place= Bonn | mr=2075634 |year= 2003}} *{{Cite arXiv |first1=Olivier |last1=Espinosa |first2=Victor Hugo |last2=Moll |author-link2=Victor Hugo Moll |title=The evaluation of Tornheim double sums |year=2008 |eprint=math/0505647 }} *{{cite journal |first1=Olivier |last1=Espinosa |first2=Victor Hugo |last2=Moll |author-link2=Victor Hugo Moll |title=The evaluation of Tornheim double sums II |journal=Ramanujan J. |year=2010 |volume=22 |pages=55–99 |mr=2610609 |doi=10.1007/s11139-009-9181-1 |arxiv=0811.0557 |s2cid=17055581 }} *{{Cite journal | last1=Borwein | first1=J.M. | author1-link=J.M. Borwein | last2=Chan | first2=O-Y. | title=Duality in tails of multiple zeta values | year=2010 |doi = 10.1142/S1793042110003058 |journal= Int. J. Number Theory |volume=6 |issue=3 |mr=2652893 |pages=501–514 | citeseerx=10.1.1.157.9158 }} *{{Cite journal|first1=Ankur | last1=Basu |title=On the evaluation of Tornheim sums and allied double sums |year=2011 |doi=10.1007/s11139-011-9302-5 |journal=Ramanujan J. |volume=26 |pages=193–207 |issue=2 |mr=2853480 | s2cid=120229489 }} ==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 --- Adapted from the Wikipedia article [Multiple zeta function](https://en.wikipedia.org/wiki/Multiple_zeta_function) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Multiple_zeta_function?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.