Lehmer's Conjecture (the Mahler Measure Problem)

For α an algebraic number, let M(α) = max(1, |\(a_{d}\)|) · Π max(1, |α_i|) be its Mahler measure. Claim: for every ε > 0 there are only finitely many irreducible integer polynomials with M(P) ≥ 1 + ε, independent of degree.

Lehmer's 1933 polynomial \(x^{10}\) + \(x^{9}\) − \(x^{7}\) − \(x^{6}\) − \(x^{5}\) − \(x^{4}\) − \(x^{3}\) + x + 1 has measure ≈ 1.17628, still the smallest known above 1. Dobrowolski (1979) proved the definitive lower bound m(α) ≥ c·(log log d / log d)^3 / d for degree d — tending to zero, so infinitely many near-Lehmer numbers may exist in principle. The conjecture drives arithmetic dynamics (distributions of preperiodic points), Bogomolov-type statements, and explicit class field theory; every advance in equidistribution quietly re-proves pieces of it.