In mathematics, the Witten zeta function, introduced by , is a function associated to a root system

Root system

In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras...

 that encodes the degrees of the irreducible representations of the corresponding Lie group

Lie group

In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...

. It is a special case of the Shintani zeta function

Shintani zeta function

In mathematics, a Shintani zeta function or Shintani L-function is a generalization of the Riemann zeta function. They were studied studied by...

.

Definition

Witten's original definition of the zeta function of a semisimple Lie group was
where the sum is over equivalence classes of irreducible representations R.

If Δ of rank r is a root system with n positive roots in Δ⁺ and with simple roots λ_i, the Witten zeta function of several variables is given by

The original zeta function studied by Witten differed from this slightly, in that all the numbers s_α are equal, and the function is multiplied by a constant.