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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Online-Zugang: | https://arxiv.org/abs/2505.09996 |
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| _version_ | 1866916012236996608 |
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| author | Dhankhar, Priya Singh, Sanjay Kumar |
| author_facet | Dhankhar, Priya Singh, Sanjay Kumar |
| contents | Let $R$ be a finite ring with unity, $ψ: R \to \mathbb{C}^\times$ be an additive character of $R$, and \( χ_0 \) be the principal multiplicative character ($i.e.$, $χ_0(x) = 1 \quad \text{for all } x \in R^\times$), then the Gauss sum is \[ G(χ_0, ψ) = \sum_{x \in R^\times} ψ(x). \] In this paper, we give an explicit formula for a more general form of the Gauss sum $G(χ_0, ψ)$. Interestingly, the formula extends the known formula of classical Ramanujan's sum to the context of finite rings. As an application, we derive the eigenvalues for a more general form of the unitary Cayley graph $\text{Cay}(R, R^{\times})$ using the formula. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_09996 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Gauss sum with principal multiplicative character Dhankhar, Priya Singh, Sanjay Kumar Combinatorics Representation Theory 20C15, 05C25, 05C50 Let $R$ be a finite ring with unity, $ψ: R \to \mathbb{C}^\times$ be an additive character of $R$, and \( χ_0 \) be the principal multiplicative character ($i.e.$, $χ_0(x) = 1 \quad \text{for all } x \in R^\times$), then the Gauss sum is \[ G(χ_0, ψ) = \sum_{x \in R^\times} ψ(x). \] In this paper, we give an explicit formula for a more general form of the Gauss sum $G(χ_0, ψ)$. Interestingly, the formula extends the known formula of classical Ramanujan's sum to the context of finite rings. As an application, we derive the eigenvalues for a more general form of the unitary Cayley graph $\text{Cay}(R, R^{\times})$ using the formula. |
| title | Gauss sum with principal multiplicative character |
| topic | Combinatorics Representation Theory 20C15, 05C25, 05C50 |
| url | https://arxiv.org/abs/2505.09996 |