Higher Mahler measure for cyclotomic polynomials and Lehmer's question

摘要

The k-higher Mahler measure of a non-zero polynomial P is the integral of log ~k{pipe}P{pipe} 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 formulas for 2- and 3-higher Mahler measure of cyclotomic polynomials.