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Multiple zeta function

Inmathematics, the multiple zeta functions are generalizations of the Riemann zeta function, defined by

${\displaystyle \zeta (s_{1},\ldots ,s_{k})=\sum _{n_{1}>n_{2}>\cdots >n_{k}>0}\ {\frac {1}{n_{1}^{s_{1}}\cdots n_{k}^{s_{k}}}}=\sum _{n_{1}>n_{2}>\cdots >n_{k}>0}\ \prod _{i=1}^{k}{\frac {1}{n_{i}^{s_{i}}}},\!}$

and converge 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 "depth" 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,

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

Definition[edit]

Multiple zeta functions arise as special cases of the multiple polylogarithms

${\displaystyle \mathrm {Li} _{s_{1},\ldots ,s_{d}}(\mu _{1},\ldots ,\mu _{d})=\sum \limits _{k_{1}>\cdots >k_{d}>0}{\frac {\mu _{1}^{k_{1}}\cdots \mu _{d}^{k_{d}}}{k_{1}^{s_{1}}\cdots k_{d}^{s_{d}}}}}$

which are generalizations of the polylogarithm functions. When all of the ${\displaystyle \mu _{i}}$ are n^th roots of unity and the ${\displaystyle s_{i}}$ are all nonnegative integers, the values of the multiple polylogarithm are called colored multiple zeta values of level ${\displaystyle n}$. In particular, when ${\displaystyle n=2}$, they are called Euler sumsoralternating multiple zeta values, and when ${\displaystyle n=1}$ they are simply called multiple zeta values. Multiple zeta values are often written

${\displaystyle \zeta (s_{1},\ldots ,s_{d})=\sum \limits _{k_{1}>\cdots >k_{d}>0}{\frac {1}{k_{1}^{s_{1}}\cdots k_{d}^{s_{d}}}}}$

and Euler sums are written

${\displaystyle \zeta (s_{1},\ldots ,s_{d};\varepsilon _{1},\ldots ,\varepsilon _{d})=\sum \limits _{k_{1}>\cdots >k_{d}>0}{\frac {\varepsilon _{1}^{k_{1}}\cdots \varepsilon ^{k_{d}}}{k_{1}^{s_{1}}\cdots k_{d}^{s_{d}}}}}$

where ${\displaystyle \varepsilon _{i}=\pm 1}$. Sometimes, authors will write a bar over an ${\displaystyle s_{i}}$ corresponding to an ${\displaystyle \varepsilon _{i}}$ equal to ${\displaystyle -1}$, so for example

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

Integral structure and identities[edit]

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

${\displaystyle \int _{0}^{x}f_{1}($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:^[2]

${\displaystyle \mathrm {Li} _{s_{1},\ldots ,s_{d}}(\mu _{1},\ldots ,\mu _{d})=\int _{0}^{1}\left({\frac {dt}{t}}\right)^{s_{1}-1}{\frac {dt}{a_{1}-t}}\cdots \left({\frac {dt}{t}}\right)^{s_{d}-1}{\frac {dt}{a_{d}-t}}}$ where ${\displaystyle a_{j}=\prod \limits _{i=1}^{j}\mu _{i}^{-1}}$ for ${\displaystyle j=1,2,\ldots ,d}$.

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

${\displaystyle \left(\int _{0}^{x}f_{1}($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 ${\displaystyle {\mathfrak {Sh}}_{n,m}=\{\sigma \in S_{m}\mid \sigma ($1)<\cdots <\sigma (n),\sigma (n+1)<\cdots <\sigma (m)\}} and ${\displaystyle S_{m}}$ is the symmetric groupon${\displaystyle m}$ symbols.

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

${\displaystyle \zeta (\mathbf {w} )=\int _{0}^{1}\mathbf {w} }$ for any ${\displaystyle \mathbf {w} \in X^{*}}$,

which, by the aforementioned integral identity, makes

${\displaystyle \zeta (a^{s_{1}-1}b\cdots a^{s_{d}-1}b)=\zeta (s_{1},\ldots ,s_{d}).}$

Then, the integral identity on products gives^[2]

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

Two parameters case[edit]

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

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

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

${\displaystyle \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 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]

${\displaystyle \zeta (s,t)=\zeta ($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)}

s	t	approximate value	explicit formulae	OEIS
2	2	0.811742425283353643637002772406	${\displaystyle {\tfrac {3}{4}}\zeta ($4)}	A197110
3	2	0.228810397603353759768746148942	${\displaystyle 3\zeta ($2)\zeta (3)-{\tfrac {11}{2}}\zeta (5)}	A258983
4	2	0.088483382454368714294327839086	${\displaystyle \left(\zeta ($3)\right)^{2}-{\tfrac {4}{3}}\zeta (6)}	A258984
5	2	0.038575124342753255505925464373	${\displaystyle 5\zeta ($2)\zeta (5)+2\zeta (3)\zeta (4)-11\zeta (7)}	A258985
6	2	0.017819740416835988362659530248		A258947
2	3	0.711566197550572432096973806086	${\displaystyle {\tfrac {9}{2}}\zeta ($5)-2\zeta (2)\zeta (3)}	A258986
3	3	0.213798868224592547099583574508	${\displaystyle {\tfrac {1}{2}}\left(\left(\zeta ($3)\right)^{2}-\zeta (6)\right)}	A258987
4	3	0.085159822534833651406806018872	${\displaystyle 17\zeta ($7)-10\zeta (2)\zeta (5)}	A258988
5	3	0.037707672984847544011304782294	${\displaystyle 5\zeta ($3)\zeta (5)-{\tfrac {147}{24}}\zeta (8)-{\tfrac {5}{2}}\zeta (6,2)}	A258982
2	4	0.674523914033968140491560608257	${\displaystyle {\tfrac {25}{12}}\zeta ($6)-\left(\zeta (3)\right)^{2}}	A258989
3	4	0.207505014615732095907807605495	${\displaystyle 10\zeta ($2)\zeta (5)+\zeta (3)\zeta (4)-18\zeta (7)}	A258990
4	4	0.083673113016495361614890436542	${\displaystyle {\tfrac {1}{2}}\left(\left(\zeta ($4)\right)^{2}-\zeta (8)\right)}	A258991

Note that if ${\displaystyle s+t=2p+2}$ we have ${\displaystyle p/3}$ irreducibles, i.e. these MZVs cannot be written as function of ${\displaystyle \zeta ($a)} only.^[5]

Three parameters case[edit]

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

${\displaystyle \zeta (a,b,c)=\sum _{n>j>i\geq 1}\ {\frac {1}{n^{a}j^{b}i^{c}}}=\sum _{n=1}^{\infty }{\frac {1}{(n+2)^{a}}}\sum _{j=1}^{n}{\frac {1}{(j+1)^{b}}}\sum _{i=1}^{j}{\frac {1}{($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[edit]

The above MZVs satisfy the Euler reflection formula:

${\displaystyle \zeta (a,b)+\zeta (b,a)=\zeta ($a)\zeta (b)-\zeta (a+b)} for ${\displaystyle a,b>1}$

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

${\displaystyle \zeta (a,b,c)+\zeta (a,c,b)+\zeta (b,a,c)+\zeta (b,c,a)+\zeta (c,a,b)+\zeta (c,b,a)=\zeta ($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 ${\displaystyle a,b,c>1}$

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

Symmetric sums in terms of the zeta function[edit]

Let ${\displaystyle S(i_{1},i_{2},\cdots ,i_{k})=\sum _{n_{1}\geq n_{2}\geq \cdots n_{k}\geq 1}{\frac {1}{n_{1}^{i_{1}}n_{2}^{i_{2}}\cdots n_{k}^{i_{k}}}}}$, and for a partition ${\displaystyle \Pi =\{P_{1},P_{2},\dots ,P_{l}\}}$ of the set ${\displaystyle \{1,2,\dots ,k\}}$, let ${\displaystyle c(\Pi )=(\left|P_{1}\right|-1)!(\left|P_{2}\right|-1)!\cdots (\left|P_{l}\right|-1)!}$. Also, given such a ${\displaystyle \Pi }$ and a k-tuple ${\displaystyle i=\{i_{1},...,i_{k}\}}$ of exponents, define ${\displaystyle \prod _{s=1}^{l}\zeta (\sum _{j\in P_{s}}i_{j})}$.

The relations between the ${\displaystyle \zeta }$ and ${\displaystyle S}$ are: ${\displaystyle S(i_{1},i_{2})=\zeta (i_{1},i_{2})+\zeta (i_{1}+i_{2})}$ and ${\displaystyle S(i_{1},i_{2},i_{3})=\zeta (i_{1},i_{2},i_{3})+\zeta (i_{1}+i_{2},i_{3})+\zeta (i_{1},i_{2}+i_{3})+\zeta (i_{1}+i_{2}+i_{3}).}$

Theorem 1 (Hoffman)[edit]

For any real ${\displaystyle i_{1},\cdots ,i_{k}>1,}$, ${\displaystyle \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 ${\displaystyle i_{j}}$ are all distinct. (There is no loss of generality, since we can take limits.) The left-hand side can be written as ${\displaystyle \sum _{\sigma }\sum _{n_{1}\geq n_{2}\geq \cdots \geq n_{k}\geq 1}{\frac {1}{{n^{i_{1}}}_{\sigma ($1)}{n^{i_{2}}}_{\sigma (2)}\cdots {n^{i_{k}}}_{\sigma (k)}}}} . Now thinking on the symmetric

group ${\displaystyle \Sigma _{k}}$ as acting on k-tuple ${\displaystyle n=(1,\cdots ,k)}$ of positive integers. A given k-tuple ${\displaystyle n=(n_{1},\cdots ,n_{k})}$ has an isotropy group

${\displaystyle \Sigma _{k}($n)} and an associated partition ${\displaystyle \Lambda }$of${\displaystyle (1,2,\cdots ,k)}$: ${\displaystyle \Lambda }$ is the set of equivalence classes of the relation given by ${\displaystyle i\sim j}$ iff ${\displaystyle n_{i}=n_{j}}$, and ${\displaystyle \Sigma _{k}($n)=\{\sigma \in \Sigma _{k}:\sigma (i)\sim \forall i\}} . Now the term ${\displaystyle {\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 ${\displaystyle \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 ${\displaystyle \left|\Sigma _{k}($n)\right|} times. It occurs on the right-hand side in those terms corresponding to partitions ${\displaystyle \Pi }$ that are refinements of ${\displaystyle \Lambda }$: letting ${\displaystyle \succeq }$ denote refinement, ${\displaystyle {\frac {1}{{n^{i_{1}}}_{\sigma ($1)}{n^{i_{2}}}_{\sigma (2)}\cdots {n^{i_{k}}}_{\sigma (k)}}}} occurs ${\displaystyle \sum _{\Pi \succeq \Lambda }(\Pi )}$ times. Thus, the conclusion will follow if ${\displaystyle \left|\Sigma _{k}($n)\right|=\sum _{\Pi \succeq \Lambda }c(\Pi )} for any k-tuple ${\displaystyle n=\{n_{1},\cdots ,n_{k}\}}$ and associated partition ${\displaystyle \Lambda }$. To see this, note that ${\displaystyle c(\Pi )}$ counts the permutations having cycle type specified by ${\displaystyle \Pi }$: since any elements of ${\displaystyle \Sigma _{k}($n)} has a unique cycle type specified by a partition that refines ${\displaystyle \Lambda }$, the result follows.^[6]

For ${\displaystyle k=3}$, the theorem says ${\displaystyle \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 ${\displaystyle i_{1},i_{2},i_{3}>1}$. This is the main result of.^[7]

Having ${\displaystyle \zeta (i_{1},i_{2},\cdots ,i_{k})=\sum _{n_{1}>n_{2}>\cdots n_{k}\geq 1}{\frac {1}{n_{1}^{i_{1}}n_{2}^{i_{2}}\cdots n_{k}^{i_{k}}}}}$. To state the analog of Theorem 1 for the ${\displaystyle \zeta 's}$, we require one bit of notation. For a partition

${\displaystyle \Pi =\{P_{1},\cdots ,P_{l}\}}$of${\displaystyle \{1,2\cdots ,k\}}$, let ${\displaystyle {\tilde {c}}(\Pi )=(-1)^{k-l}c(\Pi )}$.

Theorem 2 (Hoffman)[edit]

For any real ${\displaystyle i_{1},\cdots ,i_{k}>1}$, ${\displaystyle \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 ${\displaystyle \sum _{\sigma }\sum _{n_{1}>n_{2}>\cdots >n_{k}\geq 1}{\frac {1}{{n^{i_{1}}}_{\sigma ($1)}{n^{i_{2}}}_{\sigma (2)}\cdots {n^{i_{k}}}_{\sigma (k)}}}} , and a term ${\displaystyle {\frac {1}{n_{1}^{i_{1}}n_{2}^{i_{2}}\cdots n_{k}^{i_{k}}}}}$ occurs on the left-hand since once if all the ${\displaystyle n_{i}}$ are distinct, and not at all otherwise. Thus, it suffices to show ${\displaystyle \sum _{\Pi \succeq \Lambda }{\tilde {c}}(\Pi )={\begin{cases}1,{\text{ if }}\left|\Lambda \right|=k\\0,{\text{ otherwise }}.\end{cases}}}$ (1)

To prove this, note first that the sign of ${\displaystyle {\tilde {c}}(\Pi )}$ is positive if the permutations of cycle type ${\displaystyle \Pi }$ 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 ${\displaystyle \Sigma _{k}($n)} . But such an isotropy group has equal numbers of even and odd permutations unless it is trivial, i.e. unless the associated partition ${\displaystyle \Lambda }$is${\displaystyle \{\{1\},\{2\},\cdots ,\{k\}\}}$.^[6]

The sum and duality conjectures^[6][edit]

We first state the sum conjecture, which is due to C. Moen.^[8]

Sum conjecture (Hoffman). For positive integers k and n, ${\displaystyle \sum _{i_{1}+\cdots +i_{k}=n,i_{1}>1}\zeta (i_{1},\cdots ,i_{k})=\zeta ($n)} , where the sum is extended over k-tuples ${\displaystyle i_{1},\cdots ,i_{k}}$ of positive integers with ${\displaystyle i_{1}>1}$.

Three remarks concerning this conjecture are in order. First, it implies ${\displaystyle \sum _{i_{1}+\cdots +i_{k}=n,i_{1}>1}S(i_{1},\cdots ,i_{k})={n-1 \choose k-1}\zeta ($n)} . Second, in the case ${\displaystyle k=2}$ it says that ${\displaystyle \zeta (n-1,1)+\zeta (n-2,2)+\cdots +\zeta (2,n-2)=\zeta ($n)} , or using the relation between the ${\displaystyle \zeta 's}$ and ${\displaystyle S's}$ and Theorem 1, ${\displaystyle 2S(n-1,1)=(n+1)\zeta ($n)-\sum _{k=2}^{n-2}\zeta (k)\zeta (n-k).}