Multiple zeta function

For a different but related multiple zeta function, see Barnes zeta function.

In mathematics, the multiple zeta functions are generalisations of the 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 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 "length" 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,

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

Two parameters case

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

\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.

Multiple zeta functions are known to satisfy what is known as MZV duality, the simplest case of which is the famous identity of 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 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]

\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	${\tfrac {3}{4}}\zeta (4)$	A197110
3	2	0.228810397603353759768746148942	$3\zeta (2)\zeta (3)-{\tfrac {11}{2}}\zeta (5)$
4	2	0.088483382454368714294327839086	$\left(\zeta (3)\right)^{2}-{\tfrac {4}{3}}\zeta (6)$
5	2	0.038575124342753255505925464373	$5\zeta (2)\zeta (5)+2\zeta (3)\zeta (4)-11\zeta (7)$
6	2	0.017819740416835988
2	3	0.711566197550572432096973806086	${\tfrac {9}{2}}\zeta (5)-2\zeta (2)\zeta (3)$
3	3	0.213798868224592547099583574508	${\tfrac {1}{2}}\left(\left(\zeta (3)\right)^{2}-\zeta (6)\right)$
4	3	0.085159822534833651406806018872	$17\zeta (7)-10\zeta (2)\zeta (5)$
5	3	0.037707672984847544011304782294	$5\zeta (3)\zeta (5)-{\tfrac {147}{24}}\zeta (8)-{\tfrac {5}{2}}\zeta (6,2)$
2	4	0.674523914033968140491560608257	${\tfrac {25}{12}}\zeta (6)-\left(\zeta (3)\right)^{2}$
3	4	0.207505014615732095907807605495	$10\zeta (2)\zeta (5)+\zeta (3)\zeta (4)-18\zeta (7)$
4	4	0.083673113016495361614890436542	${\tfrac {1}{2}}\left(\left(\zeta (4)\right)^{2}-\zeta (8)\right)$

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.^[5]

Three parameters case

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

\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_{{i,c}}}{(j+1)^{b}}}

Euler reflection formula

The above MZVs satisfy the Euler 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 that:^[5]

\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 $i_{1},\cdots ,i_{k}>1,$ , $\sum _{{{\sigma \in \sum _{{k}}}}}S(i_{{\sigma (1)}},\dots ,i_{{\sigma (k)}})=\sum _{{{\text{partitions }}\Pi {\text{ of }}\{1,\dots ,k\}}}c(\Pi )\zeta (i,\Pi )$ .

Proof. Assume the $i_j$ are all distinct. (There is not loss of generality, since we can take limits.) The left-hand side can be written as $\sum _{{\sigma }}\sum _{{n_{1}\geq n_{2}\geq \cdots \geq n_{k}\geq 1}}{\frac {1}{{n^{{i_{1}}}}_{{\sigma (1)}}{n^{{i_{2}}}}_{{\sigma (2)}}\cdots {n^{{i_{k}}}}_{{\sigma (k)}}}}$ . Now thinking on the symmetric

group $\sum _{{k}}$ 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 group

$\sum _{{k}}(n)$ and an associated partition $\Lambda$ of $(1,2,\cdots ,k)$ : $\Lambda$ is the set of equivalence classes of the relation given by $i\sim j$ iff $n_{i}=n_{j}$ , and $\sum _{{k}}(n)=\{\sigma \in \sum _{{k}}:\sigma (i)\sim \forall i\}$ . 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 $\sum _{{{\sigma \in \sum _{{k}}}}}S(i_{{\sigma (1)}},\dots ,i_{{\sigma (k)}})=\sum _{{{\text{partitions }}\Pi {\text{ of }}\{1,\dots ,k\}}}c(\Pi )\zeta (i,\Pi )$ exactly $\left|\sum _{{k}}(n)\right|$ 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 $\left|\sum _{{k}}(n)\right|=\sum _{{\Pi \succeq \Lambda }}c(\Pi )$ 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 specified by $\Pi$ : since any elements of $\sum _{{k}}(n)$ has a unique cycle-type specified by a partition that refines $\Lambda$ , the result follows.^[6]

For $k=3$ , the theorem says $\sum _{{{\sigma \in \sum _{{3}}}}}S(i_{{\sigma (1)}},i_{{\sigma (2)}},i_{{\sigma (3)}})=\zeta (i_{1})\zeta (i_{2})\zeta (i_{3})+\zeta (i_{1}+i_{2})\zeta (i_{3})+\zeta (i_{1})\zeta (i_{2}+i_{3})+\zeta (i_{1}+i_{3})\zeta (i_{2})+2\zeta (i_{1}+i_{2}+i_{3})$ for $i_{1},i_{2},i_{3}>1$ . This is the main result of.^[7]

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

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

Theorem 2(Hoffman)

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

Proof. We follow the same line of argument as in the preceding proof. The left-hand side is now $\sum _{{\sigma }}\sum _{{n_{1}>n_{2}>\cdots >n_{k}\geq 1}}{\frac {1}{{n^{{i_{1}}}}_{{\sigma (1)}}{n^{{i_{2}}}}_{{\sigma (2)}}\cdots {n^{{i_{k}}}}_{{\sigma (k)}}}}$ , and a term ${\frac {1}{n_{{1}}^{{i_{1}}}n_{{2}}^{{i_{2}}}\cdots n_{{k}}^{{i_{k}}}}}$ occurs on the left-hand since once if all the $n_{i}$ 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, 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 $\sum _{{k}}(n)$ . But such an isotropy group has equal numbers of even and odd permutations unless it is trivial, i.e. unless the associated partition $\Lambda$ is $\{\{1\},\{2\},\cdots ,\{k\}\}$ .^[6]

The sum and duality conjectures^[6]

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

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 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's paper^[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 $\tau$ on the set $\Im$ of finite sequences of positive integers whose first element is greater than 1. Let $\mathrm {T}$ be the set of strictly increasing finite sequences of positive integers, and let $\Sigma :\Im \rightarrow \mathrm{T}$ be the function that sends a sequence in $\Im$ to its sequence of partial sums. If $\mathrm{T} _{n}$ is the set of sequences in $Tau$ whose last element is at most $n$ , we have two commuting involutions $R_{n}$ and $C_{n}$ on $\mathrm{T} _{n}$ defined by $R_{n}(a_{1},a_{2},\cdots ,a_{l})=(n+1-a_{l},n+1-a_{{l-1}},\cdots ,n+1-a_{1})$ and $C_{n}(a_{1},\cdots ,a_{l})$ = complement of $\{a_{1},\cdots ,a_{l}\}$ in $\{1,2,\cdots ,n\}$ arranged in increasing order. The our definition of $\tau$ is $\tau (I)=\Sigma ^{{-1}}R_{n}C_{n}\Sigma (I)=\Sigma ^{{-1}}C_{n}R_{n}\Sigma (I)$ for $I=(i_{1},i_{2},\cdots ,i_{k})\in \Im$ with $i_{1}+\cdots +i_{k}=n$ .

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

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

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]

\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.^[5]

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.

\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 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)

if $a=b$ 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:^[5]

\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 whom the argument of the factorials is $\geqslant 0$

Other results

For any integer positive : $a,b,\dots ,k$ :

\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 Matsumoto (2003) who was motivated by the papers Mordell (1958) and 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.

References

Tornheim, Leonard (1950). "Harmonic double series". American Journal of Mathematics. 72: 303–314. doi:10.2307/2372034. ISSN 0002-9327. MR 0034860.
Mordell, Louis J. (1958). "On the evaluation of some multiple series". Journal of the London Mathematical Society. Second Series. 33: 368–371. doi:10.1112/jlms/s1-33.3.368. ISSN 0024-6107. MR 0100181.
Apostol, Tom M.; Vu, Thiennu H. (1984), "Dirichlet series related to the Riemann zeta function", Journal of Number Theory, 19 (1): 85–102, doi:10.1016/0022-314X(84)90094-5, ISSN 0022-314X, MR 0751166
Crandall, Richard E.; Buhler, Joe P. (1994). "On the evaluation of Euler Sums". Experimental Mathematics. 3 (4): 275. doi:10.1080/10586458.1994.10504297. MR 1341720.
Borwein, Jonathan M.; Girgensohn, Roland (1996). "Evaluation of Triple Euler Sums". El. J. Combinat. 3 (1): #R23. MR 1401442.
Flajolet, Philippe; Salvy, Bruno (1998). "Euler Sums and contour integral representations". Exp. Math. 7.
Zhao, Jianqiang (1999). "Analytic continuation of multiple zeta functions". Proceedings of the American Mathematical Society. 128 (5): 1275–1283. doi:10.1090/S0002-9939-99-05398-8. MR 1670846.
Matsumoto, Kohji (2003), "On Mordell–Tornheim and other multiple zeta-functions", Proceedings of the Session in Analytic Number Theory and Diophantine Equations, Bonner Math. Schriften, 360, Bonn: Univ. Bonn, MR 2075634
Espinosa, Olivier; Moll, Victor H. (2008). "The evaluation of Tornheim double sums". arXiv:0811.0557.
Espinosa, Olivier; Moll, Victor H. (2010). "The evaluation of Tornheim double sums II". Ramanujan J. 22: 55–99. doi:10.1007/s11139-009-9181-1. MR 2610609.
Borwein, J.M.; Chan, O-Y. (2010). "Duality in tails of multiple zeta values". Int. J. Number Theory. 6 (3): 501–514. doi:10.1142/S1793042110003058. MR 2652893.
Basu, Ankur (2011). "On the evaluation of Tornheim sums and allied double sums". Ramanujan J. 26 (2): 193–207. doi:10.1007/s11139-011-9302-5. MR 2853480.

Notes

↑ Zhao, Jianqiang (2010). "Standard relations of multiple polylogarithm values at roots of unity". Documenta Mathematica. 15: 1–34.
↑ Zhao, Jianqiang (2016). Multiple Zeta Functions, Multiple Polylogarithms and Their Special Values. World Scientific Publishing. ISBN 978-981-4689-39-7.
1 2 Hoffman, Mike. "Multiple Zeta Values". Mike Hoffman's Home Page. U.S. Naval Academy. Retrieved June 8, 2012.
1 2 Borwein, David; Borwein, Jonathan; Bradley, David (September 23, 2004). "Parametric Euler Sum Identities" (PDF). CARMA, AMSI Honours Course. The University of Newcastle. Retrieved June 3, 2012.
1 2 3 4 Broadhurst, D. J. (1996). "On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory.". arXiv:hep-th/9604128.
1 2 3 4 Hoffman, Michael (1992). "Multiple Harmonic Series". Pacific Journal of Mathematics. 152: 276–278. doi:10.2140/pjm.1992.152.275. MR 1141796. Zbl 0763.11037.
↑ Ramachandra Rao, R. Sita; M. V. Subbarao (1984). "Transformation formulae for multiple series". Pacific Journal of Mathematics. 113: 417–479. doi:10.2140/pjm.1984.113.471.
1 2 Moen, C. "Sums of Simple Series". Preprint.
↑ Euler, L. (1775). "Meditationes circa singulare serierum genus". Novi Comm. Acad. Sci. Petropol. 15 (20): 140–186.
↑ Williams, G. T. (1958). "On the evaluation of some multiple series". Journal of the London Mathematical Society. 33: 368–371. doi:10.1112/jlms/s1-33.3.368.

External links

Borwein, Jonathan; Zudilin, Wadim. "Lecture notes on the Multiple Zeta Function".
Hoffman, Michael (2012). "Multiple zeta values".
Zhao, Jianqiang (2016). Multiple Zeta Functions, Multiple Polylogarithms and Their Special Values. World Scientific Publishing. ISBN 978-981-4689-39-7.

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Multiple zeta function

Two parameters case

Three parameters case

Euler reflection formula

Symmetric sums in terms of the zeta function

Theorem 1(Hoffman)

Theorem 2(Hoffman)

The sum and duality conjectures[6]

Euler sum with all possible alternations of sign

Notation

Reflection formula

Other relations

Other results

Mordell–Tornheim zeta values

References

Notes

External links

The sum and duality conjectures^[6]