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(si) > 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, ..., sk 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 + ... + sk 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 Hn 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)
stapproximate valueexplicit formulaeOEIS
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=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.

There 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 increasig 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

Notes

  1. ↑ 1.0 1.1 Hoffman, Mike. "Multiple Zeta Values". Mike Hoffman's Home Page. U.S. Naval Academy. Retrieved June 8, 2012.
  2. ↑ 2.0 2.1 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. ↑ 3.0 3.1 3.2 3.3 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. ↑ 4.0 4.1 4.2 4.3 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. ↑ 6.0 6.1 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.

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