Witten zeta function

If ${\displaystyle G}$ is a compact semisimple Lie group, the associated Witten zeta function is (the meromorphic continuation of) the series

${\displaystyle \zeta _{G}(s)=\sum _{\rho }{\frac {1}{(\dim \rho )^{s}}},}$

where the sum is over equivalence classes of irreducible representations of ${\displaystyle G}$.

In the case where ${\displaystyle G}$ is connected and simply connected, the correspondence between representations of ${\displaystyle G}$ and of its Lie algebra, together with the Weyl dimension formula, implies that ${\displaystyle \zeta _{G}(s)}$ can be written as

${\displaystyle \sum _{m_{1},\dots ,m_{r}>0}\prod _{\alpha \in \Phi ^{+}}{\frac {1}{\langle \alpha ^{\lor },m_{1}\lambda _{1}+\cdots +m_{r}\lambda _{r}\rangle ^{s}}},}$

where ${\displaystyle \Phi ^{+}}$ denotes the set of positive roots, ${\displaystyle \{\lambda _{i}\}}$ is a set of simple roots and ${\displaystyle r}$ is the rank.

• ${\displaystyle \zeta _{SU(2)}(s)=\zeta (s)}$, the Riemann zeta function.
• ${\displaystyle \zeta _{SU(3)}(s)=\sum _{x=1}^{\infty }\sum _{y=1}^{\infty }{\frac {1}{(xy(x+y)/2)^{s}}}.}$

If ${\displaystyle G}$ is simple and simply connected, the abscissa of convergence of ${\displaystyle \zeta _{G}(s)}$ is ${\displaystyle r/\kappa }$, where ${\displaystyle r}$ is the rank and ${\displaystyle \kappa =|\Phi ^{+}|}$. This is a theorem due to Alex Lubotzky and Michael Larsen.[3] A new proof is given by Jokke Häsä and Alexander Stasinski in.[4] The proof in [4] yields a more general result, namely it gives an explicit value (in terms of simple combinatorics) of the abscissa of convergence of any "Mellin zeta function" of the form

${\displaystyle \sum _{x_{1},\dots ,x_{r}=1}^{\infty }{\frac {1}{P(x_{1},\dots ,x_{r})^{s}}},}$

where ${\displaystyle P(x_{1},\dots ,x_{r})}$ is a product of linear polynomials with non-negative real coefficients.