Approximations to q-logarithms and q-dilogarithms, with applications to q-zeta values

We construct simultaneous rational approximations to q-series L₁(x₁; q) and L1(x₂; q) and, if x = x₁ = x₂, to series L₁(x; q) and L₂(x; q), where [formula could not be replicated], [formula could not be replicated]. Applying the construction, we obtain quantitative linear independence over ℚ of the numbers in the following collections: 1, ζ_q(1) = L₁(1; q), ζ_q+00B2(1) and 1, ζ_q(1), ζ_q(2) = L₂(1; q) for q = 1/p, p ε ℤ {0,±1}.