Linear independence of $q$-analogue of the generalized Stieltjes constants over number fields
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866916205104726016 |
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| author | Chatterjee, Tapas Garg, Sonam |
| author_facet | Chatterjee, Tapas Garg, Sonam |
| contents | In this article, we aim to extend the research conducted by Chatterjee and Garg in 2024, particularly focusing on the $q$-analogue of the generalized Stieltjes constants. These constants constitute the coefficients in the Laurent series expansion of a $q$-analogue of the Hurwitz zeta function around $s=1$. Chatterjee and Garg previously established arithmetic results related to $γ_0(q,x)$, for $q>1$ and $0 < x <1$ over the field of rational numbers. Here, we broaden their findings to encompass number fields $\mathbb{F}$ in two scenarios: firstly, when $\mathbb{F}$ is linearly disjoint from the cyclotomic field $\mathbb{Q}(ζ_b)$, and secondly, when $\mathbb{F}$ has non-trivial intersection with $\mathbb{Q}(ζ_b)$, with $b \geq 3$ being any positive integer. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_09139 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Linear independence of $q$-analogue of the generalized Stieltjes constants over number fields Chatterjee, Tapas Garg, Sonam Number Theory In this article, we aim to extend the research conducted by Chatterjee and Garg in 2024, particularly focusing on the $q$-analogue of the generalized Stieltjes constants. These constants constitute the coefficients in the Laurent series expansion of a $q$-analogue of the Hurwitz zeta function around $s=1$. Chatterjee and Garg previously established arithmetic results related to $γ_0(q,x)$, for $q>1$ and $0 < x <1$ over the field of rational numbers. Here, we broaden their findings to encompass number fields $\mathbb{F}$ in two scenarios: firstly, when $\mathbb{F}$ is linearly disjoint from the cyclotomic field $\mathbb{Q}(ζ_b)$, and secondly, when $\mathbb{F}$ has non-trivial intersection with $\mathbb{Q}(ζ_b)$, with $b \geq 3$ being any positive integer. |
| title | Linear independence of $q$-analogue of the generalized Stieltjes constants over number fields |
| topic | Number Theory |
| url | https://arxiv.org/abs/2404.09139 |