Multiple zeta function

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 sums or alternating 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

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

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 group on ${\displaystyle m}$ 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]

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

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	0.017819740416835988		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]

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

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

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.

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 ${\displaystyle S(i_{1},i_{2},i_{3})=\zeta (i_{1},i_{2},i_{3})+\zeta (i_{1}+i_{2},i_{3})+\zeta (i_{1},i_{2}+i_{3})+\zeta (i_{1}+i_{2}+i_{3}).}$

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, ${\displaystyle 2S(n-1,1)=(n+1)\zeta (n)-\sum _{k=2}^{n-2}\zeta (k)\zeta (n-k).}$

This was proved by Euler[9] and has been rediscovered several times, in particular by Williams.[10] Finally, C. Moen[8] has proved the same conjecture for k=3 by lengthy but elementary arguments. For the duality conjecture, we first define an involution ${\displaystyle \tau }$ on the set ${\displaystyle \Im }$ of finite sequences of positive integers whose first element is greater than 1. Let ${\displaystyle \mathrm {T} }$ be the set of strictly increasing finite sequences of positive integers, and let ${\displaystyle \Sigma$ :\Im \rightarrow \mathrm {T} } be the function that sends a sequence in ${\displaystyle \Im }$ to its sequence of partial sums. If ${\displaystyle \mathrm {T} _{n}}$ is the set of sequences in ${\displaystyle \mathrm {T} }$ whose last element is at most ${\displaystyle n}$, we have two commuting involutions ${\displaystyle R_{n}}$ and ${\displaystyle C_{n}}$ on ${\displaystyle \mathrm {T} _{n}}$ defined by ${\displaystyle R_{n}(a_{1},a_{2},\dots ,a_{l})=(n+1-a_{l},n+1-a_{l-1},\dots ,n+1-a_{1})}$ and ${\displaystyle C_{n}(a_{1},\dots ,a_{l})}$ = complement of ${\displaystyle \{a_{1},\dots ,a_{l}\}}$ in ${\displaystyle \{1,2,\dots ,n\}}$ arranged in increasing order. The our definition of ${\displaystyle \tau }$ is ${\displaystyle \tau (I)=\Sigma ^{-1}R_{n}C_{n}\Sigma (I)=\Sigma ^{-1}C_{n}R_{n}\Sigma (I)}$ for ${\displaystyle I=(i_{1},i_{2},\dots ,i_{k})\in \Im }$ with ${\displaystyle i_{1}+\cdots +i_{k}=n}$.

For example, ${\displaystyle \tau (3,4,1)=\Sigma ^{-1}C_{8}R_{8}(3,7,8)=\Sigma ^{-1}(3,4,5,7,8)=(3,1,1,2,1).}$ We shall say the sequences ${\displaystyle (i_{1},\dots ,i_{k})}$ and ${\displaystyle \tau (i_{1},\dots ,i_{k})}$ are dual to each other, and refer to a sequence fixed by ${\displaystyle \tau }$ as self-dual.[6]

Duality conjecture (Hoffman). If ${\displaystyle (h_{1},\dots ,h_{n-k})}$ is dual to ${\displaystyle (i_{1},\dots ,i_{k})}$, then ${\displaystyle \zeta (h_{1},\dots ,h_{n-k})=\zeta (i_{1},\dots ,i_{k})}$.

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 s₁ > 1) MZVs of the partitions of length k and weight n, with 1 ≤ k ≤ n − 1. In formula:[3]

${\displaystyle \sum _{\stackrel {s_{1}+\cdots +s_{k}=n}{s_{1}>1}}\zeta (s_{1},\ldots ,s_{k})=\zeta (n).}$

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

${\displaystyle \zeta (6,1)+\zeta (5,2)+\zeta (4,3)+\zeta (3,4)+\zeta (2,5)=\zeta (7).}$

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

For all positive integers ${\displaystyle a,b,\dots ,k}$:

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

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

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

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

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

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

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

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

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

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

The Mordell–Tornheim zeta function, introduced by Matsumoto (2003) who was motivated by the papers Mordell (1958) and Tornheim (1950), is defined by

${\displaystyle \zeta _{MT,r}(s_{1},\dots ,s_{r};s_{r+1})=\sum _{m_{1},\dots ,m_{r}>0}{\frac {1}{m_{1}^{s_{1}}\cdots m_{r}^{s_{r}}(m_{1}+\dots +m_{r})^{s_{r+1}}}}}$

It is a special case of the Shintani zeta function.

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