Copyright © 2010 Peter Bundschuh and Keijo Väänänen. This is an open access article distributed under the
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For fixed complex with , the -logarithm is the meromorphic continuation of the series , into the whole complex plane. If is an algebraic number field, one may ask if are linearly independent over for satisfying . In 2004, Tachiya showed that this is true in the Subcase , , , and the present authors extended this result to arbitrary integer from an imaginary quadratic number field , 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 is the Eisenstein number field , an integer from , and a primitive third root of unity. Under these conditions, the linear independence holds also for , and both results are quantitative.