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

Volume 180 / 2017

Dan Romik 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.

Authors

  • Dan RomikDepartment of Mathematics
    University of California, Davis
    One Shields Ave.
    Davis, CA 95616, U.S.A.
    e-mail

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