## On the number of $n$-dimensional representations of $\operatorname{SU}(3)$, the Bernoulli numbers, and the Witten zeta function

### Volume 180 / 2017

Acta Arithmetica 180 (2017), 111-159
MSC: Primary 11P82; Secondary 11B68.
DOI: 10.4064/aa8455-3-2017
Published online: 30 August 2017

#### Abstract

We derive new results about properties of the Witten zeta function associated with the group ${\rm SU }(3)$, and use them to prove an asymptotic formula for the number of $n$-dimensional representations of ${\rm SU }(3)$ counted up to equivalence. Our analysis also relates the Witten zeta function of ${\rm SU} (3)$ to a summation identity for Bernoulli numbers discovered in 2008 by Agoh and Dilcher. We give a new proof of that identity and show that it is a special case of a stronger identity involving the Eisenstein series.