Higher Mahler measure for cyclotomic polynomials and Lehmer's question (Q411285)

Let \(P \in \mathbb{C}[x]\) be a nonzero polynomial and let \(k \in \mathbb{N}\). The \(k\)-higher (logarithmic) Mahler measure of \(P\) is defined by \[ m_k(P):= \frac{1}{2\pi i} \int_{|x|=1} \log^k|P(x)| \frac{dx}{x}. \] In their main result (Theorem 4) the authors show that for each integer \(s \geq 1\) and each \(P \in \mathbb{Z}[x]\) which is not a monomial we have \(m_{2s}(P) \geq (\pi^2/12)^s\) if \(P\) is reciprocal and \(m_{2s}(P) \geq (\pi^2/48)^s\) if \(P\) is non-reciprocal. Another result (Theorem 6) is devoted to the evaluation of \(m_3(P)\) for a polynomial \(P \in \mathbb{C}[x]\) with all roots on the unit circle. The paper also contains a number of other interesting results on this \(k\)-higher generalization of Mahler's measure.