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In mathematics, the multiple zeta functions 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, ..., si are all positive integers these sums are often called multiple zeta values (MZVs) or Euler sums.

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,

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

and

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

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:

$$\zeta(2,2k-1)=\sum_{j=1}^{k-2}2j\zeta(2j+1)\zeta(2k-2j)+(2k-1)\zeta(2k-1)\zeta(2)-\frac{2k^2+k+1}2\zeta(2k+1) for natural k\ge 2$$

s1 s2 approximate value explicit formulae
2 2 0.811742425283353643637002772406 $$\tfrac{3}{4}\zeta(4)$$
2 3 0.228810397603353759768746148942 $$3\zeta(2)\zeta(3)-\tfrac{11}{2}\zeta(5)$$
2 4 0.088483382454368714294327839086 $$\left (\zeta(3)\right )^2-\tfrac{4}{3}\zeta(6)$$
2 5 0.038575124342753255505925464373 $$5\zeta(2)\zeta(5)+2\zeta(3)\zeta(4)-11\zeta(7)$$
3 2 0.711566197550572432096973806086 $$\tfrac{9}{2}\zeta(5)-2\zeta(2)\zeta(3)$$
3 3 0.213798868224592547099583574508 $$\tfrac{1}{2}\left (\left (\zeta(3)\right )^2 -\zeta(6)\right )$$
3 4 0.085159822534833651406806018872 $$17\zeta(7)-10\zeta(2)\zeta(5)$$
3 5 0.037707672984847544011304782294
4 2 0.674523914033968140491560608257 $$\tfrac{25}{12}\zeta(6)-\left (\zeta(3)\right )^2$$
4 3 0.207505014615732095907807605495 $$10\zeta(2)\zeta(5)+\zeta(3)\zeta(4)-18\zeta(7)$$
4 4 0.083673113016495361614890436542 $$\tfrac{1}{2}\left (\left (\zeta(4)\right )^2 -\zeta(8)\right )$$

These MZVs satisfy the relation:

$$\zeta(a,b)+\zeta(b,a)=\zeta(a)\zeta(b)-\zeta(a+b) for a,b>1$$

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. MR0034860.
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. MR0100181.
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, MR0751166
Crandall, Richard E.; Buhler, Joe P. (1994). "On the evaluation of Euler Sums". Experimental Mathematics 3 (4): 275. MR1341720.
Borwein, Jonathan M.; Girgensohn, Roland (1996). "Evaluation of Triple Euler Sums". El. J. Combinat. 3 (1): #R23. MR1401442.
Zhao, Jianqiang (1999). "Analytic continuation of multiple zeta functions". Proceedings of American Mathematics Society 128 (5): 1275–1283. doi:10.1090/S0002-9939-99-05398-8. MR1670846.
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, MR2075634
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. MR2652893.

Borwein, Jonathan; Zudilin, Wadim. "Lecture notes on the Multiple Zeta Function".
Hoffman, Michael (2012). "Multiple zeta values".