Multiple 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.

The k in the above definition is named the "length" of a MZV, and the n = s₁ + ... + s_k is known as the "weight".^[1]

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):^[2]

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

$\zeta(s,t)=\sum_{n=1}^\infty \frac{H_{n,t}}{(n+1)^s}$ 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):^[2]

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

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.^[3]

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:^[3]

$\zeta(a,b,c)+\zeta(a,c,b)+\zeta(b,a,c)+\zeta(b,c,a)+\zeta(c,a,b)+\zeta(c,b,a)=\zeta(a)\zeta(b)\zeta(c)+2\zeta(a+b+c)-\zeta(a)\zeta(b+c)-\zeta(b)\zeta(a+c)-\zeta(c)\zeta(a+b)$ 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 $S(i_1,i_2,\cdots,i_k)=\sum_{n_1\geq n_2\geq\cdots n_k\geq1}\frac{1}{n_1^{i_1} n_2^{i_2}\cdots n_k^{i_k}}$, 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 $\sum_{\sigma}\sum_{n_1\geq n_2 \geq \cdots \geq n_k \geq1} \frac{1}{{n^{i_1}}_{\sigma(1)}{n^{i_2}}_{\sigma(2)} \cdots {n^{i_k}}_{\sigma(k)} }$. 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.^[4]

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.^[5]

Having $\zeta(i_1,i_2,\cdots,i_k)=\sum_{n_1> n_2>\cdots n_k\geq1}\frac{1}{n_1^{i_1} n_2^{i_2}\cdots n_k^{i_k}}$. 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 $\sum_{\sigma}\sum_{n_1 > n_2 > \cdots > n_k \geq1} \frac{1}{{n^{i_1}}_{\sigma(1)}{n^{i_2}}_{\sigma(2)} \cdots {n^{i_k}}_{\sigma(k)} }$, and a term $\frac{1}{n^{i_1}_{1}n^{i_2}_{2} \cdots n^{i_k}_{k}}$ occurs on the left-hand since once if all the $n_i$ are distinct, and not at all otherwise. Thus, it suffices to show $\sum_{\Pi\succeq\Lambda}\tilde{c}(\Pi)=\begin{cases} 1,\text{ if } \left| \Lambda \right|=k \\ 0, \text{ otherwise }. \end{cases}$ (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 $\{\{1\},\{2\},\cdots,\{k\}\}$.^[4]

The sum and duality conjectures^[4]

We first state the sum conjecture, which is due to C. Moen.^[6]

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^[7] and has been rediscovered several times, in particular by Williams.^[8] Finally, C. Moen^[6] 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 $\Tau$ be the set of strictly increasing finite sequences of positive integers, and let $\Sigma : \Im \rightarrow \Tau$ be the function that sends a sequence in $\Im$ to its sequence of partial sums. If $\Tau_n$ 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 $\Tau_n$ 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 $\tau(I)=\Sigma^{-1}R_nC_n\Sigma(I)=\Sigma^{-1}C_nR_n\Sigma(I)$ for $I=(i_1,i_2,\cdots,i_k) \in \Im$ with $i_1+\cdots+i_k=n$.

For example, $\tau(3,4,1)=\Sigma^{-1}C_8R_8(3,7,8)=\Sigma^{-1}(3,4,5,7,8)=(3,1,1,2,1).$ 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.^[4]

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:^[1]

$\sum_\stackrel{s_1 + \cdots + s_k=n}{s_1>1}\zeta(s_1, \ldots, s_k) = \zeta(n)$

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.^[3]

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:^[3]

$\zeta(a,b)+\zeta(\bar{a},\bar{b})=\sum_{s>0} (a+b-s-1)!\Big[\frac{Z_a(a+b-s,s)}{(a-s)!(b-1)!}+\frac{Z_b(a+b-s,s)}{(b-s)!(a-1)!}\Big]$

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 $\geqslant0$

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

1. Hoffman, Mike. "Multiple Zeta Values". Mike Hoffman's Home Page. U.S. Naval Academy. Retrieved June 8, 2012.
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.
3. 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.
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.
5. 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.
6. Moen, C. "Sums of Simple Series". Preprint.
7. Euler, L. (1775). "Meditationes circa singulare serierum genus". Novi Comm. Acad. Sci. Petropol 15 (20): 140–186.
8. 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".