Linear Independence of ð ‘ž -Logarithms over the Eisenstein Integers

For fixed complex ð ‘ž with | ð ‘ž | > 1 , the ð ‘ž -logarithm ð ¿ ð ‘ž is the meromorphic continuation of the series ∑ ð ‘› > 0 𠑧 ð ‘› / ( ð ‘ž ð ‘› − 1 ) , | 𠑧 | < | ð ‘ž | , into the whole complex plane. If ð ¾ is an algebraic number field, one may ask if 1 , ð ¿ ð ‘ž ( 1 ) , ð ¿ ð ‘ž ( ð ‘ ) are linearly independent over ð ¾ for ð ‘ž , ð ‘ âˆˆ ð ¾ Ã— satisfying | ð ‘ž | > 1 , ð ‘ â‰ ð ‘ž , ð ‘ž 2 , ð ‘ž 3 , … . In 2004, Tachiya showed that this is true in the Subcase ð ¾ = â„š , ð ‘ž ∈ ℤ , ð ‘ = − 1 , and the present authors extended this result to arbitrary integer ð ‘ž from an imaginary quadratic number field ð ¾ , 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 ð ¾ is the Eisenstein number field √ â„š ( − 3 ) , ð ‘ž an integer from ð ¾ , and ð ‘ a primitive third root of unity. Under these conditions, the linear independence holds also for 1 , ð ¿ ð ‘ž ( ð ‘ ) , ð ¿ ð ‘ž ( ð ‘ âˆ’ 1 ) , and both results are quantitative.