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

Abstract

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 $1/4<\theta <1/3$. In this paper, we obtain an asymptotic formula for the number of irreducible representations of $\mathfrak{so}(5)$. More precisely, the irreducible representations of the Lie algebra $\mathfrak{so}(5)$ are a family of representations of dimension $jk(j+k)(j+2k)/6$ for $j,k\in {\mathbb{N}}_0$ and suppose that ${\varrho }_{\mathfrak{so}(5)}(n)$ is the number of irreducible $\mathfrak{so}(5)$ representations of dimension $n$. We obtain an asymptotic formula for the summatory function ${\sum }_{1\le n\le x}{\varrho }_{\mathfrak{so}(5)}(n)$.

All comments are welcome. This is the first draft of the paper