On the number of irreducible representations of \(\mathfrak{su}(3)\) (Q6606886)

The irreducible representations of the Lie algebra \(\mathfrak{su}(3)\) are a family of representations \(W_{j,k}\) of dimension \(\frac{jk(j+k)}{2}\) for \(j,k\in\mathbb{N}_0\). In the paper under review, the authors study the average of the number of irreducible \(\mathfrak{su}(3)\)-representations of dimension \(n\), given by\N\[\N\varrho(n) = \sum_{\substack{j,k \geq 1 \\\N\frac{jk(j+k)}{2} = n}} 1.\N\]\NInspired by the similarity of \(\varrho(n)\) to the divisor function, the authors use a variant of the hyperbola method to prove that\N\[\N\sum_{1\le n\le x} \varrho(n) = \frac{2^\frac23\sqrt3\Gamma(1/3)^3}{4\pi}x^\frac23 + 2^\frac32\zeta(1/2)\sqrt{x} + O(x^\frac13).\N\]\NMeanwhile, as a bi-product, the authors deduce from the proof of the above result, the following integral computation\N\[\N\int_0^\frac12 \frac{2t^\frac52-t^{-\frac12}}{\sqrt{1+t^3}} dt = \frac34 - \frac{2^\frac23\sqrt3\Gamma(1/3)^3}{8\pi},\N\]\Nwhich can be compared with \(\frac{39}{4}-6 \sqrt{2} \, _2F_1\left(-\frac{5}{6},\frac{1}{2};\frac{1}{6};-\frac{1}{8}\right)\), a hypergeometric representation output for the above integral by Wolfram Mathematica software.