Linear independence of \(q\)-logarithms over the Eisenstein integers (Q978998)

Summary: For fixed complex \(q\) with \(| q| >1\), the \(q\)-logarithm \(L_q\) is the meromorphic continuation of the series \(\sum_{n>0} z^n/(q^n-1)\), \(| z|<| q|\), into the whole complex plane. If \(K\) is an algebraic number field, one may ask if \(1, L_q(1), L_q(c)\) are linearly independent over\(K\) for \(q,c\in K^\times\) satisfying \(| q| >1\), \(c\neq q, q^2, q^3,\ldots\). In 2004, \textit{Y. Tachiya} [Tokyo J. Math. 27, 75--85 (2004; Zbl 1072.11051)] showed that this is true in the subcase \(K=\mathbb Q\), \(q\in\mathbb Z\), \(c=-1\), and the present authors extended this result to arbitrary integer \(q\) from an imaginary quadratic number field \(K\), and provided a quantitative version. In this paper, the earlier method, in particular its arithmetical part, is further developed to answer the above question in the affirmative if \(K\) is the Eisenstein number field \(\mathbb Q(\sqrt{-3})\), \(q\) an integer from \(K\), and \(c\) a primitive third root of unity. Under these conditions, the linear independence holds also for \(1, L_q(c), L_q(c^{-1})\), and both results are quantitative.