Sequences related to Lehmer's problem

The Mahler measure of a monic polynomial $P(x)=a_d{x}^d+a_{d-1}{x}^{d-1}+\cdots +a_{1}x+a_0$ is defined as $M(P):=\mathrm{\mid }{a}_d\mathrm{\mid }{\prod }_{P(\alpha )=0}\max \{1,\mathrm{\mid }\alpha \mathrm{\mid }\}$, where the product runs over all roots of $P$. Lehmer’s problem asks whether there exists a constant $C>1$ such that $M(P)\ge C$ 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.