Higher Mahler measure for cyclotomic polynomials and Lehmer's question

Abstract

The $k$-higher Mahler measure of a nonzero polynomial $P$ is the integral of ${\log }^k|P|$ on the unit circle. In this note, we consider Lehmer's question (which is a long-standing open problem for $k=1$) for $k>1$ and find some interesting formulae for 2- and 3-higher Mahler measure of cyclotomic polynomials.