On linear independence of logarithms of certain rational numbers (Q908954)

Let \(\Phi_ d(x)\) be the cyclotomic polynomial satisfying \((1-x^ r)=\prod_{d | r}\Phi_ d(x)\). The author proves the following results. Suppose that \(r\in {\mathbb{N}}\), \(a,b\in {\mathbb{Q}}\) and \(a\neq 0\), \(| b| >\gamma | a|^{r+1}\), \(\gamma =12(2r\prod_{p| r}p^{1/(p-1)})^ r\), then the numbers 1, ln \(\Phi_ 1(a/b),...,\ln \Phi_ d(a/b),...,\ln \Phi_ r(a/b)\) (d \(| r)\) are linearly independent over \({\mathbb{Q}}\). Further the lower estimate of linear forms of 1, ln \(\Phi_ 1(a/b),...,\ln \Phi_ d(a/b),...,\ln \Phi_ r(a/b)\) (d \(| r)\) is given.