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.

The k in the above definition is named the "length" of a MZV, and the n = s₁ + ... + s_k is known as the "weight".

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):

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

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

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

s	t	approximate value	explicit formulae	OEIS
2	2	0.811742425283353643637002772406	${\tfrac {3}{4}}\zeta (4)$	OEIS: A197110
3	2	0.228810397603353759768746148942	$3\zeta (2)\zeta (3)-{\tfrac {11}{2}}\zeta (5)$	OEIS: A258983
4	2	0.088483382454368714294327839086	$\left(\zeta (3)\right)^{2}-{\tfrac {4}{3}}\zeta (6)$	OEIS: A258984
5	2	0.038575124342753255505925464373	$5\zeta (2)\zeta (5)+2\zeta (3)\zeta (4)-11\zeta (7)$	OEIS: A258985
6	2	0.017819740416835988		OEIS: A258947
2	3	0.711566197550572432096973806086	${\tfrac {9}{2}}\zeta (5)-2\zeta (2)\zeta (3)$	OEIS: A258986
3	3	0.213798868224592547099583574508	${\tfrac {1}{2}}\left(\left(\zeta (3)\right)^{2}-\zeta (6)\right)$	A258987
4	3	0.085159822534833651406806018872	$17\zeta (7)-10\zeta (2)\zeta (5)$	A258988
5	3	0.037707672984847544011304782294	$5\zeta (3)\zeta (5)-{\tfrac {147}{24}}\zeta (8)-{\tfrac {5}{2}}\zeta (6,2)$	A258982
2	4	0.674523914033968140491560608257	${\tfrac {25}{12}}\zeta (6)-\left(\zeta (3)\right)^{2}$	A258989
3	4	0.207505014615732095907807605495	$10\zeta (2)\zeta (5)+\zeta (3)\zeta (4)-18\zeta (7)$	A258990
4	4	0.083673113016495361614890436542	${\tfrac {1}{2}}\left(\left(\zeta (4)\right)^{2}-\zeta (8)\right)$	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 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:

$\zeta (a,b,c)+\zeta (a,c,b)+\zeta (b,a,c)+\zeta (b,c,a)+\zeta (c,a,b)+\zeta (c,b,a)=\zeta (a)\zeta (b)\zeta (c)+2\zeta (a+b+c)-\zeta (a)\zeta (b+c)-\zeta (b)\zeta (a+c)-\zeta (c)\zeta (a+b)$ 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.

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.

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

The sum and duality conjectures

We first state the sum conjecture, which is due to C. Moen.

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 and has been rediscovered several times, in particular by Williams. 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 $\tau$ 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.

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:

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

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.

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

$\zeta (a,b)+\zeta ({\bar {a}},{\bar {b}})=\sum _{s>0}(a+b-s-1)!{\Big [}{\frac {Z_{a}(a+b-s,s)}{(a-s)!(b-1)!}}+{\frac {Z_{b}(a+b-s,s)}{(b-s)!(a-1)!}}{\Big ]}$

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 (2): 303–314. doi:10.2307/2372034. ISSN 0002-9327. JSTOR 2372034. MR 0034860.
• Mordell, Louis J. (1958). "On the evaluation of some multiple series". Journal of the London Mathematical Society. Second Series. 33 (3): 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.
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• 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, vol. 360, Bonn: Univ. Bonn, MR 2075634
• Espinosa, Olivier; Moll, Victor H. (2008). "The evaluation of Tornheim double sums". arXiv:0811.0557 [math.NT].
• 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. CiteSeerX 10.1.1.157.9158. 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.

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. Series on Number Theory and its Applications. Vol. 12. World Scientific Publishing. doi:10.1142/9634. ISBN 978-981-4689-39-7.
• Burgos Gil, José Ignacio; Fresán, Javier. "Multiple zeta values: from numbers to motives" (PDF).