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

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 γ₀(q,x), for $q>1$ and $0 < x <1$ over the field of rational numbers. Here, we broaden their findings to encompass number fields F in two scenarios: firstly, when F is linearly disjoint from the cyclotomic field Q(ζb), and secondly, when F has non-trivial intersection with Q(ζb), with b ≥ 3 being any positive integer.