The mean values of logarithms of algebraic integers (Q1292621)

Let \(p >1\). For an algebraic integer \(\alpha\) of degree \(d\) which is not a root of unity, let \(M_p(\alpha)\) be the \(p\)-th root of the mean value \[ {1\over d}\sum_{i=1}^d\bigl| \log| \alpha_i| \bigr| ^p, \] \(\alpha_1=\alpha,\dots,\alpha_d\) being the conjugates of \(\alpha\). The author obtains lower bounds for \(M_p(\alpha)\) of the form \[ M_p(\alpha)>{1\over d}(b_p-\epsilon)\delta(d), \] valid for \(d>d(\epsilon)\), with \(b_p\) being an explicit, but rather complicated constant, and \(\delta(d)=(\log\log d/\log d)^3\). A lower bound is also given for the \(p\)-th root of the mean value of \(p\)-th powers of absolute values of conjugates of \(\alpha\).