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