Linear Independence of q‐Logarithms over the Eisenstein Integers
Peter Bundschuh, Keijo Väänänen · International Journal of Mathematics and Mathematical Sciences · 2010
For fixed complex q with |q | > 1, the q‐logarithm Lq is the meromorphic continuation of the series ∑n>0zn/(qn − 1), |z | 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 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.