A multiple zeta value (MZV) of length *k* and weight *n* is a
*k*-fold infinite series of the form

where the sum is over

The MZV of length 1 and weight *n* is just the
value of the Riemann zeta function
at *n*, i.e. the harmonic series of exponent *n*.
MZVs are also known as multiple harmonic series. They occur in connection with
Kontsevich's multiple integral defining an invariant of knots and links, and
Drinfeld's work on quantum groups. They also appear in quantum field theory.

MZVs satisfy many striking relations; perhaps the simplest is (2,1) = (3). This is the first instance of the following result.

This result was proved in the case *k*=2
by L. Euler in 1775.
A proof of the case *k*=3 appeared in 1996 (*Journal of Number
Theory*, vol. 60, 329-331). Shortly afterward brief and elegant proofs
of the general case were found by Andrew Granville (see the
references for his proof) and by Don Zagier.
Another striking identity is the duality theorem, which can be stated as
follows. For any sequence

there is a dual sequence (of the same weight)

The duality theorem states that ; it
follows easily from a representation of MZVs as iterated integrals.
A third family of identities is given by the derivation theorem:
let *D*
send, for example, the sequence (2,1,3) to (3,1,3) + (2,2,3) + (2,1,4).
If we extend to sums of sequences
in the obvious way, we have the following result.

A remarkable result of Yasuo Ohno (see the references) includes the sum, duality, and derivation theorems as special cases.

Yet another identity, not included in Ohno's result, is

where there are *n* blocks of 3,1 on the left and 2*n* 2's
on the right (so both sides have weight 4*n*). This was conjectured
by Don Zagier and proved by David Broadhurst. In fact, this appears to be
just the simplest of a whole family of similar
identities: see the paper
"Combinatorial aspects of multiple zeta values" for details. Another
family of identities, the cyclic sum theorem,
was proved in 2000 by Ohno. For more details see the
paper "Relations of multiple zeta values
and their algebraic expression".

MZVs satisfy many more relations, and their global structure is not fully understood. See the talk "Algebraic structures on the set of multiple zeta values" for one approach to this problem. The basis conjecture, proposed in 1997, has recently been established through the work of Francis Brown and Zagier (whose papers can be found here and here respectively).

MZVs can be generalized further by introducing powers of ±1, or even arbitrary roots of unity, in the numerators. The resulting series have been called multiple polylogarithms at roots of unity, or (more compactly) Euler sums. For details see the talk "Algebras of multiple zeta values, quasi-symmetric functions, and Euler sums" or the paper "Special values of multidimensional polylogarithms" by Borwein, Bradley, Broadhurst and Lisonek.

You can play around with multiple zeta values yourself using the
EZFace
calculator at CECM (Centre for Experimental and Constructive Mathematics
at Simon Fraser University). The calculator gives numerical
values of MZVs with up to 100 decimal places accuracy; to find the
numerical value
of (3,2,1) you type in `z(3,2,1)`.
The calculator also "understands" expressions like `Pi^2/6`.
Here's a suggestive set of entries:

z(2,1,3)-z(2,2,2)-2*z(3,3) z(2,1,2,3)-z(2,2,2,2)-2*z(2,3,3) z(2,1,2,2,3)-z(2,2,2,2,2)-2*z(2,2,3,3) z(2,1,2,2,2,3)-z(2,2,2,2,2,2)-2*z(2,2,2,3,3) z(2,1,2,2,2,2,3)-z(2,2,2,2,2,2,2)-2*z(2,2,2,2,3,3)(They are all known to be exactly zero. The general identity resisted proof for a long time, but was finally disposed of by M. Hirose and N. Sato in 2017. See the paper "Hoffman's conjectural identity"). The calculator also has a function to look for relations of linear dependence;

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