Lehmer's Mahler Measure Problem — The Smallest Value Above 1

For an integer polynomial $P(x)=a_n\prod(x-\alpha_i)$, the Mahler measure is $M(P)=|a_n|\prod\max(1,|\alpha_i|)$. Kronecker's theorem says that an integer polynomial of measure 1 is, up to sign and monomial factors, a product of cyclotomic polynomials. Lehmer (1933) asked whether measures above 1 are uniformly separated from 1.

Lehmer's own candidate (1933) is $L(x) = x^{10} + x^9 - x^7 - x^6 - x^5 - x^4 - x^3 + x + 1$ with $M(L) \approx 1.17628$. For 90 years no smaller value has been found. A positive gap would tie integer-polynomial roots into a spectrum with deep connections to number theory and dynamics.

Smyth (1971) proved $M(P)\ge\theta_0\approx1.3247$ for non-reciprocal polynomials. Dobrowolski (1979) proved the general asymptotic bound $\log M(P)\ge c(\log\log d/\log d)^3$. Because it still tends to zero with the degree, it does not supply Lehmer's uniform gap. Mahler measure has deep links to short geodesics in arithmetic hyperbolic manifolds, but not the simple equivalence previously claimed here.