Linear Independence of q-Logarithms over the Eisenstein Integers

For fixed complex q with |q| > 1, the q-logarithm Lq is the meromorphic continuation of the series ∑ n>0 z / q −1 , |z| < |q|, into the whole complex plane. IfK is an algebraic number field, one may ask if 1, Lq 1 , Lq c are linearly independent over K for q, c ∈ K× satisfying |q| > 1, c / q, q2, q3, . . .. In 2004, Tachiya showed that this is true in the Subcase K Q, q ∈ 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 Q √−3 , q an integer from K, and c a primitive third root of unity. Under these conditions, the linear independence holds also for 1, Lq c , Lq c−1 , and both results are quantitative.