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 k-tuples of positive integers, and are positive integers that add up to n with .
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 2n 2's on the right (so both sides have weight 4n). 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; lindep([a,b,c]) looks for a vanishing linear combination of a,b,c with integer coefficients. This makes it easy (EZ?) to discover new identities!