Linear independence of values of logarithms revisited

Abstract: Let $m\ge 2$ be an integer, $K$ an algebraic number field and $\alpha\in K\setminus {0,-1}$ with sufficiently small absolute value. In this article, we provide a new lower bound for linear form in $1,{\rm{log}}(1+\alpha),\ldots,{\rm{log}}^{m-1}(1+\alpha)$ with algebraic integer coefficients in both complex and $p$-adic cases (see Theorem $2.1$ and Theorem $2.4$). Especially, in the complex case, our result is a refinement of the result of Nesterenko-Waldschmidt on the lower bound of linear form in certain values of power of logarithms. The main integrant is based on Hermite-Mahler's Pad\'{e} approximation of exponential and logarithm functions.