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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2509.17873 |
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| _version_ | 1866916960910966784 |
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| author | Nayak, Saudamini Meher, Nabin Kumar Rout, Sudhansu Sekhar |
| author_facet | Nayak, Saudamini Meher, Nabin Kumar Rout, Sudhansu Sekhar |
| contents | Let $d(n)$ be the divisor function and it is well known that $\sum_{1\leq n \leq x}d(n) = x\log x+(2γ-1)x +\mathcal{O}\left(x^{θ+ε}\right)$ where $γ$ is the Euler constant, $ε>0$ and $1/4<θ<1/3$. 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 $\sum_{1\leq n \leq x}\varrho_{\mathfrak{so}(5)}(n)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_17873 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the number of irreducible representations of $\so(5)$ Nayak, Saudamini Meher, Nabin Kumar Rout, Sudhansu Sekhar Number Theory 17B15, 11N45, 11N56 Let $d(n)$ be the divisor function and it is well known that $\sum_{1\leq n \leq x}d(n) = x\log x+(2γ-1)x +\mathcal{O}\left(x^{θ+ε}\right)$ where $γ$ is the Euler constant, $ε>0$ and $1/4<θ<1/3$. 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 $\sum_{1\leq n \leq x}\varrho_{\mathfrak{so}(5)}(n)$. |
| title | On the number of irreducible representations of $\so(5)$ |
| topic | Number Theory 17B15, 11N45, 11N56 |
| url | https://arxiv.org/abs/2509.17873 |