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arXiv · 2404.09139

Linear independence of $q$-analogue of the generalized Stieltjes constants over number fields

Tapas Chatterjee·Sonam Garg
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Abstract

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 $\gamma_0(q,x)$, 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 $\mathbb{Q}(\zeta_b)$, and secondly, when $\mathbb{F}$ has non-trivial intersection with $\mathbb{Q}(\zeta_b)$, with $b \geq 3$ being any positive integer.

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Tapas Chatterjee, Sonam Garg. 2024-04-11. Linear independence of $q$-analogue of the generalized Stieltjes constants over number fields. https://arxiv.org/abs/2404.09139

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