Lehmer's problem on the Mahler measure

For a nonzero polynomial $P(x)=a_d\prod_{i=1}^d(x-\alpha_i)\in\mathbb C[x]$, with roots listed with multiplicity, the Mahler measure is

$$M(P)=|a_d|\prod_{i=1}^d\max\bigl(1,|\alpha_i|\bigr).$$

For an algebraic number $\alpha$, set $M(\alpha)=M(P_\alpha)$, where $P_\alpha\in\mathbb Z[x]$ is the minimal polynomial of $\alpha$ with coprime coefficients and positive leading coefficient. If $P\in\mathbb Z[x]$ is nonzero, then $M(P)\ge1$. By Kronecker's theorem, an algebraic integer $\alpha\ne0$ has $M(\alpha)=1$ exactly when $\alpha$ is a root of unity.

Lehmer's problem. Does there exist an absolute constant $c>0$ such that

$$M(\alpha)\ge1+c$$

for every nonzero algebraic integer $\alpha$ that is not a root of unity?

Equivalently: is there $\mu>1$ such that every $P\in\mathbb Z[x]$ with $M(P)>1$ satisfies $M(P)\ge\mu$? Lehmer's conjecture asserts that the answer is yes. A solution must settle the question, either by proving such a $c$ exists or by producing algebraic integers $\alpha_k$, none a root of unity, with $M(\alpha_k)\to1$. The smallest known value above $1$ is attained by Lehmer's polynomial

$$L(x)=x^{10}+x^9-x^7-x^6-x^5-x^4-x^3+x+1,\qquad M(L)=1.17628081\ldots ,$$

and a stronger form of the conjecture says that $M(L)$ is the optimal value of $1+c$. [1, 6, 7]

Background

Lehmer raised the question in 1933 while generalizing Mersenne numbers [1]. For a monic integer polynomial $P$ with roots $\alpha_i$, he studied the integers $\Delta_n=\prod_i(\alpha_i^n-1)$. These grow roughly like $M(P)^n$, so polynomials with small measure give slowly growing sequences that are more likely to contain primes. He found $L(x)$ in this search and asked whether measures can come arbitrarily close to $1$.

The problem is now central in the theory of heights. The quantity $\log M(\alpha)/\deg\alpha$ is the absolute logarithmic Weil height of $\alpha$, so Lehmer's problem asks for a lower bound on the height of order $1/\deg\alpha$. Small Mahler measures are also tied to Salem numbers, since $M(L)$ is a Salem number, and to dynamics: the topological entropy of an automorphism of a torus equals the logarithm of the Mahler measure of its characteristic polynomial. Smyth's survey treats these connections [6].