Transcendence of generalized Euler–Lehmer constants

Linear and algebraic independence of generalized Euler–Briggs constants

“…This result is new and we will give a proof of this result in the next section. A particular case of Theorem 12 was noticed in [6]. In the same paper, the authors also proved the following theorem which will be required to prove Theorem 9.…”

“…The last equality has been established by Diamond and Ford in [4]. In a recent work, the first and the third author along with Sneh Bala Sinha [6] (see also [7]) proved the following results; Then the set T := {γ(Ω, a, q) | Ω ∈ U } is infinite and has at most one algebraic element.…”

“…In order to prove these theorems, one needs to find a closed formula for generalized Euler-Briggs constants. In [6], it was proved that γ(Ω, a, q) = 1 ϕ(q)…”

“…The articles [6,7] can be thought of a generalization of the work of R. Murty and Zaytseva [17] on generalized Euler constants. Whereas the results of R. Murty and Zaytseva are obtained by Hermite and Lindemann theorem, the proofs in [6] require careful analysis of units in cyclotomic fields and Baker's theorem on linear forms in logarithms.…”

“…One possible approach is to consider γ not in isolation, but as an element of the infinite family of generalised Euler-Briggs constants. In a recent work [6], it is shown that the infinite list of generalized Euler-Briggs constants can have at most one algebraic number. In this paper, we study the dimension of spaces generated by these generalized Euler-Briggs constants over number fields.…”