Sequences related to Lehmer's problem

TL;DR

This paper explores integer sequences connected to Lehmer's problem, aiming to understand their properties and potential implications for resolving whether a universal Mahler measure lower bound exists for noncyclotomic polynomials.

Contribution

It investigates properties of specific integer sequences related to Lehmer's problem, providing new insights into their structure and potential role in solving the problem.

Findings

Identified properties of sequences linked to Mahler measures

Analyzed how these sequences relate to Lehmer's conjecture

Suggested directions for future research on sequence-based approaches

Abstract

The Mahler measure of a monic polynomial $P(x) = a_dx^d + a_{d-1}x^{d-1} + \dots + a_1x + a_0$ is defined as $M(P) := |a_d| \prod_{P(\alpha)=0} \max\{1, |\alpha|\}$, where the product runs over all roots of $P$. Lehmer's problem asks whether there exists a constant $C>1$ such that $M(P) \geq C$ for all noncyclotomic polynomials in $\mathbb{Z}[x]$. In this thesis, we examine the properties of various integer sequences related to this problem, with special focus on how these sequences might help solving Lehmer's problem.