On the number of irreducible representations of $\text{so}(5)$

Let $d(n)$ be the divisor function and it is well known that ${\sum }_{1\le n\le x}d(n)=x\log x+(2\gamma -1)x+\mathcal{O}\left(x^{\theta +\epsilon }\right)$ where $\gamma$ is the Euler constant, $\epsilon >0$ and $1\mathrm{/}4<\theta <1\mathrm{/}3$. In this paper, we obtain an asymptotic formula for the number of irreducible representations of $\mathfrak{s}\mathfrak{o}(5)$. More precisely, the irreducible representations of the Lie algebra $\mathfrak{s}\mathfrak{o}(5)$ are a family of representations of dimension $jk(j+k)(j+2k)\mathrm{/}6$ for $j,k\in {\mathbb{N}}_0$ and suppose that ${\varrho }_{\mathfrak{s}\mathfrak{o}(5)}(n)$ is the number of irreducible $\mathfrak{s}\mathfrak{o}(5)$ representations of dimension $n$. We obtain an asymptotic formula for the summatory function ${\sum }_{1\le n\le x}{\varrho }_{\mathfrak{s}\mathfrak{o}(5)}(n)$.