Higher Mahler measure for cyclotomic polynomials and Lehmer's question

TL;DR

This paper explores higher Mahler measures of cyclotomic polynomials for $k>1$, providing new formulas and insights related to Lehmer's question, a longstanding open problem in number theory.

Contribution

It introduces formulas for 2- and 3-higher Mahler measures of cyclotomic polynomials and extends the investigation of Lehmer's question beyond the classical case.

Findings

Derived explicit formulas for 2- and 3-higher Mahler measures.

Connected higher Mahler measures to Lehmer's question.

Provided new perspectives on Mahler measures for cyclotomic polynomials.

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.