multiple zeta function

Multiple zeta functions arise as special cases of the multiple polylogarithms

:$\backslash mathrm\{Li\}\_\{s\_1,\backslash ldots,s\_d\}(\backslash mu\_1,\backslash ldots,\backslash mu\_d)\; =\; \backslash sum\backslash limits\_\{k\_1>\backslash cdots>k\_d>0\}\backslash frac\{\backslash mu\_1^\{k\_1\}\backslash cdots\backslash mu\_d^\{k\_d\}\}\{k\_1^\{s\_1\}\backslash cdots\; k\_d^\{s\_d\}\}$

which are generalizations of the polylogarithm functions. When all of the $\backslash mu\_i$ are n^th 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

:$\backslash zeta(s\_1,\backslash ldots,s\_d)\; =\; \backslash sum\backslash limits\_\{k\_1\; >\; \backslash cdots\; >\; k\_d\; >\; 0\}\; \backslash frac\{1\}\{k\_1^\{s\_1\}\backslash cdots\; k\_d^\{s\_d\}\}$

and Euler sums are written

:$\backslash zeta(s\_1,\backslash ldots,s\_d;\backslash varepsilon\_1,\backslash ldots,\backslash varepsilon\_d)\; =\; \backslash sum\backslash limits\_\{k\_1\; >\; \backslash cdots\; >\; k\_d\; >\; 0\}\; \backslash frac\{\backslash varepsilon\_1^\{k\_1\}\backslash cdots\; \backslash varepsilon^\{k\_d\}\}\{k\_1^\{s\_1\}\backslash cdots\; k\_d^\{s\_d\}\}$

where $\backslash varepsilon\_i\; =\; \backslash pm\; 1$. Sometimes, authors will write a bar over an $s\_i$ corresponding to an $\backslash varepsilon\_i$ equal to $-1$, so for example

:$\backslash zeta(\backslash overline\{a\},b)\; =\; \backslash zeta(a,b;-1,1)$.

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

:$\backslash int\_0^x\; f\_1(t)\; dt\; \backslash cdots\; f\_d(t)\; dt\; =\; \backslash int\_0^x\; f\_1(t\_1)\backslash left(\backslash int\_0^\{t\_1\}f\_2(t\_2)\backslash left(\backslash int\_0^\{t\_2\}\; \backslash cdots\; \backslash left(\; \backslash int\_0^\{t\_d\}\; f\_d(t\_d)\; dt\_d\backslash right)\backslash right)dt\_2\backslash right)dt\_1$

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

:$\backslash mathrm\{Li\}\_\{s\_1,\backslash ldots,s\_d\}(\backslash mu\_1,\backslash ldots,\backslash mu\_d)\; =\; \backslash int\_0^1\; \backslash left(\backslash frac\{dt\}\{t\}\backslash right)^\{s\_1-1\}\backslash frac\{dt\}\{a\_1-t\}\backslash cdots\; \backslash left(\backslash frac\{dt\}\{t\}\backslash right)^\{s\_d-1\}\; \backslash frac\{dt\}\{a\_d-t\}$ where $a\_j\; =\; \backslash prod\backslash limits\_\{i=1\}^j\; \backslash mu\_i^\{-1\}$ for $j\; =\; 1,2,\backslash ldots,d$.

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

:$\backslash left(\backslash int\_0^x\; f\_1(t)dt\; \backslash cdots\; f\_n(t)\; dt\; \backslash right)\backslash !\backslash left(\backslash int\_0^x\; f\_\{n+1\}(t)dt\backslash cdots\; f\_m(t)\; dt\; \backslash right)\; =$

\sum\limits_{\sigma \in \mathfrak{Sh}_{n,m}}\int_0^x f_{\sigma(1)}(t)\cdots f_{\sigma(m)}(t) where and $S\_m$ is the symmetric group on $m$ symbols.

To utilize this in the context of multiple zeta values, define $X\; =\; \backslash \{a,b\backslash \}$, $X^*$ to be the free monoid generated by $X$ and $\backslash mathfrak\{A\}$ to be the free $\backslash Q$-vector space generated by $X^*$. $\backslash mathfrak\{A\}$ 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 $a\; =\; \backslash frac\{dt\}\{t\}$, $b\; =\; \backslash frac\{dt\}\{1-t\}$, and define

:$\backslash zeta(\backslash mathbf\{w\})\; =\; \backslash int\_0^1\; \backslash mathbf\{w\}$ for any $\backslash mathbf\{w\}\; \backslash in\; X^*$,

which, by the aforementioned integral identity, makes

:$\backslash zeta(a^\{s\_1-1\}b\backslash cdots\; a^\{s\_d-1\}b)\; =\; \backslash zeta(s\_1,\backslash ldots,s\_d).$

Then, the integral identity on products gives

:$\backslash zeta(w)\backslash zeta(v)\; =\; \backslash zeta(w\; \backslash text\{\; ⧢\; \}\; v).$

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}}

:$\backslash zeta(s,t)\; =\; \backslash sum\_\{n\; >\; m\; \backslash geq\; 1\}\; \backslash \; \backslash frac\{1\}\{n^\{s\}\; m^\{t\}\}\; =\; \backslash sum\_\{n=2\}^\{\backslash infty\}\; \backslash frac\{1\}\{n^\{s\}\}\; \backslash sum\_\{m=1\}^\{n-1\}\; \backslash frac\{1\}\{m^t\}\; =\; \backslash sum\_\{n=1\}^\{\backslash infty\}\; \backslash frac\{1\}\{(n+1)^\{s\}\}\; \backslash sum\_\{m=1\}^\{n\}\; \backslash frac\{1\}\{m^t\}$

:$\backslash zeta(s,t)\; =\; \backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{H\_\{n,t\}\}\{(n+1)^s\}$ where $H\_\{n,t\}$ 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:

:$\backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{H\_n\}\{(n+1)^2\}\; =\; \backslash zeta(2,1)\; =\; \backslash zeta(3)\; =\; \backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{1\}\{n^3\},\backslash !$

where H_n 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):

:$\backslash zeta(s,t)\; =\; \backslash zeta(s)\backslash zeta(t)\; +\; \backslash tfrac\{1\}\{2\}\backslash Big[\backslash tbinom\{s+t\}\{s\}-1\backslash Big]\backslash zeta(s+t)\; -\; \backslash sum\_\{r=1\}^\{N-1\}\backslash Big[\backslash tbinom\{2r\}\{s-1\}+\backslash tbinom\{2r\}\{t-1\}\backslash Big]\backslash zeta(2r+1)\backslash zeta(s+t-1-2r)$

Note that if $s+t=2p+2$ we have $p/3$ irreducibles, i.e. these MZVs cannot be written as function of $\backslash zeta(a)$ only.

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

:$\backslash zeta(a,b,c)\; =\; \backslash sum\_\{n\; >\; j\; >\; i\; \backslash geq\; 1\}\backslash \; \backslash frac\{1\}\{n^\{a\}\; j^\{b\}\; i^\{c\}\}\; =\; \backslash sum\_\{n=1\}^\{\backslash infty\}\; \backslash frac\{1\}\{(n+2)^\{a\}\}\; \backslash sum\_\{j=1\}^n\; \backslash frac\{1\}\{(j+1)^b\}\; \backslash sum\_\{i=1\}^\{j\}\; \backslash frac\{1\}\{(i)^c\}\; =\; \backslash sum\_\{n=1\}^\{\backslash infty\}\; \backslash frac\{1\}\{(n+2)^\{a\}\}\; \backslash sum\_\{j=1\}^n\; \backslash frac\{H\_\{j,c\}\}\{(j+1)^b\}$

The above MZVs satisfy the Euler reflection formula:

:$\backslash zeta(a,b)+\backslash zeta(b,a)=\backslash zeta(a)\backslash zeta(b)-\backslash zeta(a+b)$ for $a,b>1$

Using the shuffle relations, it is easy to prove 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 }}

:$\backslash zeta(a,b,c)+\backslash zeta(a,c,b)+\backslash zeta(b,a,c)+\backslash zeta(b,c,a)+\backslash zeta(c,a,b)+\backslash zeta(c,b,a)=\backslash zeta(a)\backslash zeta(b)\backslash zeta(c)+2\backslash zeta(a+b+c)-\backslash zeta(a)\backslash zeta(b+c)-\backslash zeta(b)\backslash zeta(a+c)-\backslash zeta(c)\backslash zeta(a+b)$ for $a,b,c>1$

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

Let $S(i\_1,i\_2,\backslash cdots,i\_k)\; =\; \backslash sum\_\{n\_1\; \backslash geq\; n\_2\; \backslash geq\backslash cdots\; n\_k\; \backslash geq\; 1\}\backslash frac\{1\}\{n\_1^\{i\_1\}\; n\_2^\{i\_2\}\backslash cdots\; n\_k^\{i\_k\}\}$, and for a partition $\backslash Pi=\backslash \{P\_1,\; P\_2,\; \backslash dots,P\_l\backslash \}$ of the set $\backslash \{1,2,\backslash dots,k\backslash \}$, let $c(\backslash Pi)\; =\; (\backslash left|P\_1\backslash right|-1)!(\backslash left|P\_2\backslash right|-1)!\backslash cdots(\backslash left|P\_l\backslash right|-1)!$. Also, given such a $\backslash Pi$ and a k-tuple $i=\backslash \{i\_1,...,i\_k\backslash \}$ of exponents, define $\backslash prod\_\{s=1\}^l\; \backslash zeta(\backslash sum\_\{j\; \backslash in\; P\_s\}\; i\_j)$.

The relations between the $\backslash zeta$ and $S$ are:

$S(i\_1,i\_2)=\backslash zeta(i\_1,i\_2)+\backslash zeta(i\_1+i\_2)$ and $S(i\_1,i\_2,i\_3)=\backslash zeta(i\_1,i\_2,i\_3)+\backslash zeta(i\_1+i\_2,i\_3)+\backslash zeta(i\_1,i\_2+i\_3)+\backslash zeta(i\_1+i\_2+i\_3).$

For any real $i\_1,\backslash cdots,i\_k\; >1,$, $\backslash sum\_\{\{\backslash sigma\; \backslash in\; \backslash Sigma\_k\}\}S(i\_\{\backslash sigma(1)\},\; \backslash dots,\; i\_\{\backslash sigma(k)\})\; =\; \backslash sum\_\{\backslash text\{partitions\; \}\; \backslash Pi\; \backslash text\{\; of\; \}\; \backslash \{1,\backslash dots,k\backslash \}\}c(\backslash Pi)\backslash zeta(i,\backslash Pi)$.

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

$\backslash sum\_\{\backslash sigma\}\backslash sum\_\{n\_1\backslash geq\; n\_2\; \backslash geq\; \backslash cdots\; \backslash geq\; n\_k\; \backslash geq1\}\; \backslash frac\{1\}\{\{n^\{i\_1\}\}\_\{\backslash sigma(1)\}\{n^\{i\_2\}\}\_\{\backslash sigma(2)\}\; \backslash cdots\; \{n^\{i\_k\}\}\_\{\backslash sigma(k)\}\; \}$. Now thinking on the symmetric

group $\backslash Sigma\_k$ as acting on k-tuple $n\; =\; (1,\backslash cdots,k)$ of positive integers. A given k-tuple $n=(n\_1,\backslash cdots,n\_k)$ has an isotropy group

$\backslash Sigma\_k(n)$ and an associated partition $\backslash Lambda$ of $(1,2,\backslash cdots,k)$: $\backslash Lambda$ is the set of equivalence classes of the relation

given by $i\; \backslash sim\; j$ iff $n\_i=n\_j$, and $\backslash Sigma\_k(n)\; =\; \backslash \{\backslash sigma\; \backslash in\; \backslash Sigma\_k\; :\; \backslash sigma(i)\; \backslash sim\; \backslash forall\; i\backslash \}$. Now the term $\backslash frac\{1\}$

{{n^{i_1}}_{\sigma(1)}{n^{i_2}}_{\sigma(2)} \cdots {n^{i_k}}_{\sigma(k)}} occurs on the left-hand side of $\backslash sum\_\{\{\backslash sigma\; \backslash in\; \backslash Sigma\_k\}\}S(i\_\{\backslash sigma(1)\},\; \backslash dots,\; i\_\{\backslash sigma(k)\})\; =\; \backslash sum\_\{\backslash text\{partitions\; \}\; \backslash Pi\; \backslash text\{\; of\; \}\; \backslash \{1,\backslash dots,k\backslash \}\}c(\backslash Pi)\backslash zeta(i,\backslash Pi)$ exactly $\backslash left|\; \backslash Sigma\_k(n)\; \backslash right|$ times. It occurs on the right-hand side in those terms corresponding to partitions $\backslash Pi$ that are refinements of $\backslash Lambda$: letting $\backslash succeq$ denote refinement, $\backslash frac\{1\}$

{{n^{i_1}}_{\sigma(1)}{n^{i_2}}_{\sigma(2)} \cdots {n^{i_k}}_{\sigma(k)}} occurs $\backslash sum\_\{\backslash Pi\backslash succeq\backslash Lambda\}(\backslash Pi)$ times. Thus, the conclusion will follow if

$\backslash left|\; \backslash Sigma\_k(n)\; \backslash right|\; =\backslash sum\_\{\backslash Pi\backslash succeq\backslash Lambda\}c(\backslash Pi)$ for any k-tuple $n=\backslash \{n\_1,\backslash cdots,n\_k\backslash \}$ and associated partition $\backslash Lambda$.

To see this, note that $c(\backslash Pi)$ counts the permutations having cycle type specified by $\backslash Pi$: since any elements of $\backslash Sigma\_k(n)$ has a unique cycle type specified by a partition that refines $\backslash Lambda$, 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 $\backslash sum\_\{\{\backslash sigma\; \backslash in\; \backslash Sigma\_3\}\}S(i\_\{\backslash sigma(1)\},i\_\{\backslash sigma(2)\},i\_\{\backslash sigma(3)\})\; =\; \backslash zeta(i\_1)\backslash zeta(i\_2)\backslash zeta(i\_3)+\backslash zeta(i\_1+i\_2)\backslash zeta(i\_3)+\backslash zeta(i\_1)\backslash zeta(i\_2+i\_3)+\backslash zeta(i\_1+i\_3)\backslash zeta(i\_2)+2\backslash zeta(i\_1+i\_2+i\_3)$

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 $\backslash zeta(i\_1,i\_2,\backslash cdots,i\_k)=\backslash sum\_\{n\_1>\; n\_2>\backslash cdots\; n\_k\backslash geq1\}\backslash frac\{1\}\{n\_1^\{i\_1\}\; n\_2^\{i\_2\}\backslash cdots\; n\_k^\{i\_k\}\}$. To state the analog of Theorem 1 for the $\backslash zeta\text{'}s$, we require one bit of notation. For a partition

$\backslash Pi\; =\; \backslash \{P\_1,\backslash cdots,P\_l\backslash \}$ of $\backslash \{1,2\backslash cdots,k\backslash \}$, let $\backslash tilde\{c\}(\backslash Pi)=(-1)^\{k-l\}c(\backslash Pi)$.

For any real $i\_1,\backslash cdots,i\_k>1$, $\backslash sum\_\{\{\backslash sigma\; \backslash in\; \backslash Sigma\_k\}\}\backslash zeta(i\_\{\backslash sigma(1)\},\; \backslash dots,\; i\_\{\backslash sigma(k)\})=\backslash sum\_\{\backslash text\{partitions\; \}\; \backslash Pi\; \backslash text\{\; of\; \}\; \backslash \{1,\backslash dots,k\backslash \}\}\backslash tilde\{c\}(\backslash Pi)\backslash zeta(i,\backslash Pi)$.

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

$\backslash sum\_\{\backslash sigma\}\backslash sum\_\{n\_1\; >\; n\_2\; >\; \backslash cdots\; >\; n\_k\; \backslash geq1\}\; \backslash frac\{1\}\{\{n^\{i\_1\}\}\_\{\backslash sigma(1)\}\{n^\{i\_2\}\}\_\{\backslash sigma(2)\}\; \backslash cdots\; \{n^\{i\_k\}\}\_\{\backslash sigma(k)\}\; \}$, and a term $\backslash frac\{1\}\{n^\{i\_1\}\_\{1\}n^\{i\_2\}\_\{2\}\; \backslash cdots\; n^\{i\_k\}\_\{k\}\}$ occurs on the left-hand since once if all the $n\_i$ are distinct, and not at all otherwise. Thus, it suffices to show

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

To prove this, note first that the sign of $\backslash tilde\{c\}(\backslash Pi)$ is positive if the permutations of cycle type $\backslash Pi$ 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 $\backslash Sigma\_k(n)$. But such an isotropy group has equal numbers of even and odd permutations unless it is trivial, i.e. unless the associated partition $\backslash Lambda$ is

$\backslash \{\backslash \{1\backslash \},\backslash \{2\backslash \},\backslash cdots,\backslash \{k\backslash \}\backslash \}$.

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,

$\backslash sum\_\{i\_1+\backslash cdots+i\_k=n,\; i\_1>1\}\backslash zeta(i\_1,\backslash cdots,i\_k)\; =\; \backslash zeta(n)$, where the sum is extended over k-tuples $i\_1,\backslash cdots,i\_k$ of positive integers with $i\_1>1$.

Three remarks concerning this conjecture are in order. First, it implies

$\backslash sum\_\{i\_1+\backslash cdots+i\_k=n,\; i\_1>1\}S(i\_1,\backslash cdots,i\_k)=\{n-1\backslash choose\; k-1\}\backslash zeta(n)$. Second, in the case $k=2$ it says that $\backslash zeta(n-1,1)+\backslash zeta(n-2,2)+\backslash cdots+\backslash zeta(2,n-2)=\backslash zeta(n)$, or using the relation between the $\backslash zeta\text{'}s$ and $S\text{'}s$ and Theorem 1, $2S(n-1,1)=(n+1)\backslash zeta(n)-\backslash sum\_\{k=2\}^\{n-2\}\backslash zeta(k)\backslash zeta(n-k).$

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 $\backslash tau$ on the set $\backslash Im$ of finite sequences of positive integers whose first element is greater than 1. Let $\backslash Tau$ be the set of strictly increasing finite sequences of positive integers, and let $\backslash Sigma\; :\; \backslash Im\; \backslash rightarrow\; \backslash Tau$ be the function that sends a sequence in $\backslash Im$ to its sequence of partial sums. If $\backslash Tau\_n$ is the set of sequences in $\backslash Tau$ whose last element is at most $n$, we have two commuting involutions $R\_n$ and $C\_n$ on $\backslash Tau\_n$ defined by

$R\_n(a\_1,a\_2,\backslash dots,a\_l)=(n+1-a\_l,n+1-a\_\{l-1\},\backslash dots,n+1-a\_1)$ and

$C\_n(a\_1,\backslash dots,a\_l)$ = complement of $\backslash \{a\_1,\backslash dots,a\_l\backslash \}$ in $\backslash \{1,2,\backslash dots,n\backslash \}$ arranged in increasing order. The our definition of $\backslash tau$ is $\backslash tau(I)\; =\; \backslash Sigma^\{-1\}R\_nC\_n\backslash Sigma(I)\; =\; \backslash Sigma^\{-1\}C\_nR\_n\backslash Sigma(I)$ for $I=(i\_1,i\_2,\backslash dots,i\_k)\; \backslash in\; \backslash Im$ with $i\_1+\backslash cdots+i\_k=n$.

For example,

$\backslash tau(3,4,1)\; =\; \backslash Sigma^\{-1\}C\_8R\_8(3,7,8)\; =\; \backslash Sigma^\{-1\}(3,4,5,7,8)\; =\; (3,1,1,2,1).$

We shall say the sequences $(i\_1,\backslash dots,i\_k)$ and $\backslash tau(i\_1,\backslash dots,i\_k)$ are dual to each other, and refer to a sequence fixed by $\backslash tau$ as self-dual.

Duality conjecture (Hoffman). If $(h\_1,\backslash dots,h\_\{n-k\})$ is dual to $(i\_1,\backslash dots,i\_k)$, then $\backslash zeta(h\_1,\backslash dots,h\_\{n-k\})\; =\; \backslash zeta(i\_1,\backslash dots,i\_k)$.

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 s₁ > 1) MZVs of the partitions of length k and weight n, with 1 ≤ k ≤ n − 1. In formula:

:$\backslash sum\_\backslash stackrel\{s\_1\; +\; \backslash cdots\; +\; s\_k=n\}\{s\_1>1\}\backslash zeta(s\_1,\; \backslash ldots,\; s\_k)\; =\; \backslash zeta(n).$

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

:$\backslash zeta(6,1)+\backslash zeta(5,2)+\backslash zeta(4,3)+\backslash zeta(3,4)+\backslash zeta(2,5)\; =\; \backslash zeta(7).$

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

:$\backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{H\_n^\{(b)\}(-1)^\{(n+1)\}\}\{(n+1)^a\}=\backslash zeta(\backslash bar\{a\},b)$ with $H\_n^\{(b)\}=+1+\backslash frac\{1\}\{2^b\}+\backslash frac\{1\}\{3^b\}+\backslash cdots$ are the generalized harmonic numbers.

:$\backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{\backslash bar\{H\}\_n^\{(b)\}\}\{(n+1)^a\}=\backslash zeta(a,\backslash bar\{b\})$ with $\backslash bar\{H\}\_n^\{(b)\}=-1+\backslash frac\{1\}\{2^b\}-\backslash frac\{1\}\{3^b\}+\backslash cdots$

:$\backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{\backslash bar\{H\}\_n^\{(b)\}(-1)^\{(n+1)\}\}\{(n+1)^a\}=\backslash zeta(\backslash bar\{a\},\backslash bar\{b\})$

:$\backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{(-1)^\{n\}\}\{(n+2)^a\}\backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{\backslash bar\{H\}\_n^\{(c)\}(-1)^\{(n+1)\}\}\{(n+1)^b\}\; =\backslash zeta(\backslash bar\{a\},\backslash bar\{b\},\backslash bar\{c\})$ with $\backslash bar\{H\}\_n^\{(c)\}=-1+\backslash frac\{1\}\{2^c\}-\backslash frac\{1\}\{3^c\}+\backslash cdots$

:$\backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{(-1)^\{n\}\}\{(n+2)^a\}\backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{H\_n^\{(c)\}\}\{(n+1)^b\}=\backslash zeta(\backslash bar\{a\},b,c)$ with $H\_n^\{(c)\}=+1+\backslash frac\{1\}\{2^c\}+\backslash frac\{1\}\{3^c\}+\backslash cdots$

:$\backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{1\}\{(n+2)^a\}\backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{H\_n^\{(c)\}(-1)^\{(n+1)\}\}\{(n+1)^b\}=\backslash zeta(a,\backslash bar\{b\},c)$

:$\backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{1\}\{(n+2)^a\}\backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{\backslash bar\{H\}\_n^\{(c)\}\}\{(n+1)^b\}=\backslash zeta(a,b,\backslash bar\{c\})$

As a variant of the Dirichlet eta function we define

:$\backslash phi(s)\; =\; \backslash frac\{1-2^\{(s-1)\}\}\; \{2^\{(s-1)\}\}\; \backslash zeta(s)$ with $s>1$

:$\backslash phi(1)\; =\; -\backslash ln\; 2$

The reflection formula $\backslash zeta(a,b)+\backslash zeta(b,a)=\backslash zeta(a)\backslash zeta(b)-\backslash zeta(a+b)$ can be generalized as follows:

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

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

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

if $a=b$ we have $\backslash zeta(\backslash bar\{a\},\backslash bar\{a\})=\backslash tfrac\{1\}\{2\}\backslash Big[\backslash phi^2(a)-\backslash zeta(2a)\backslash Big]$

Using the series definition it is easy to prove:

:$\backslash zeta(a,b)+\backslash zeta(a,\backslash bar\{b\})+\backslash zeta(\backslash bar\{a\},b)+\backslash zeta(\backslash bar\{a\},\backslash bar\{b\})=\backslash frac\{\backslash zeta(a,b)\}\{2^\{(a+b-2)\}\}$ with $a>1$

:$\backslash zeta(a,b,c)+\backslash zeta(a,b,\backslash bar\{c\})+\backslash zeta(a,\backslash bar\{b\},c)+\backslash zeta(\backslash bar\{a\},b,c)+\backslash zeta(a,\backslash bar\{b\},\backslash bar\{c\})+\backslash zeta(\backslash bar\{a\},b,\backslash bar\{c\})+\backslash zeta(\backslash bar\{a\},\backslash bar\{b\},c)+\backslash zeta(\backslash bar\{a\},\backslash bar\{b\},\backslash bar\{c\})=\backslash frac\{\backslash zeta(a,b,c)\}\{2^\{(a+b+c-3)\}\}$ with $a>1$

A further useful relation is:

:$\backslash zeta(a,b)+\backslash zeta(\backslash bar\{a\},\backslash bar\{b\})=\backslash sum\_\{s>0\}\; (a+b-s-1)!\backslash Big[\backslash frac\{Z\_a(a+b-s,s)\}\{(a-s)!(b-1)!\}+\backslash frac\{Z\_b(a+b-s,s)\}\{(b-s)!(a-1)!\}\backslash Big]$

where $Z\_a(s,t)=\backslash zeta(s,t)+\backslash zeta(\backslash bar\{s\},t)-\backslash frac\{\backslash Big[\backslash zeta(s,t)+\backslash zeta(s+t)\backslash Big]\}\{2^\{(s-1)\}\}$ and $Z\_b(s,t)=\backslash frac\{\backslash zeta(s,t)\}\{2^\{(s-1)\}\}$

Note that $s$ must be used for all value $>1$ for which the argument of the factorials is $\backslash geqslant0$

For all positive integers $a,b,\backslash dots,k$:

:$\backslash sum\_\{n=2\}^\{\backslash infty\}\; \backslash zeta(n,k)\; =\; \backslash zeta(k+1)$ or more generally:

:$\backslash sum\_\{n=2\}^\{\backslash infty\}\; \backslash zeta(n,a,b,\backslash dots,k)\; =\; \backslash zeta(a+1,b,\backslash dots,k)$

:$\backslash sum\_\{n=2\}^\{\backslash infty\}\; \backslash zeta(n,\backslash bar\{k\})\; =\; -\backslash phi(k+1)$

:$\backslash sum\_\{n=2\}^\{\backslash infty\}\; \backslash zeta(n,\backslash bar\{a\},b)\; =\; \backslash zeta(\backslash overline\{a+1\},b)$

:$\backslash sum\_\{n=2\}^\{\backslash infty\}\; \backslash zeta(n,a,\backslash bar\{b\})\; =\; \backslash zeta(a+1,\backslash bar\{b\})$

:$\backslash sum\_\{n=2\}^\{\backslash infty\}\; \backslash zeta(n,\backslash bar\{a\},\backslash bar\{b\})\; =\; \backslash zeta(\backslash overline\{a+1\},\backslash bar\{b\})$

:$\backslash lim\_\{k\; \backslash to\; \backslash infty\}\backslash zeta(n,k)\; =\; \backslash zeta(n)-1$

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

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

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

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

:$\backslash zeta\_\{MT,r\}(s\_1,\backslash dots,s\_r;s\_\{r+1\})=\backslash sum\_\{m\_1,\backslash dots,m\_r>0\}\backslash frac\{1\}\{\; m\_1^\{s\_1\}\backslash cdots\; m\_r^\{s\_r\}(m\_1+\backslash dots+m\_r)^\{s\_\{r+1\}\}\}$

It is a special case of the Shintani zeta function.

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{{Reflist}}

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Category:Zeta and L-functions