nLab multiple zeta values

Idea

A multiple zeta value (MZV) is like the value of a zeta function in more than one variable.

Where zeta functions appear in physics as expressions for vacuum amplitudes, so multiple zeta functions appear in expressions for more general scattering amplitudes. The intricate combinatorics of these becomes often more tractable when re-expressing them as motivic multiple zeta values.

Definition

The standard multiple zeta values are the values of the multiple zeta function $\zeta$ of $k$ variables $s_k$ defined by

$$\zeta(s_1, \cdots , s_k) = \underset{n_1 \gt n_2 \gt \cdots \gt n_k\gt 0}{\sum} n_1^{-s_1} \cdots n_k^{-s_k} \,.$$

For $k = 1$, hence for a single argument, this reduces to the Riemann zeta function

Related concepts

• zeta function

• motivic multiple zeta values

• polylogarithm

References

• Leila Schneps, Survey of the theory of multiple zeta values (2011) (pdf)