Linear Independence of q-Logarithms over the Eisenstein Integers

Abstract EN

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.