On the Conjecture of Lehmer, Limit Mahler Measure of Trinomials and Asymptotic Expansions

Abstract

Abstract Let n ≥ 2 be an integer and denote by θn the real root in (0, 1) of the trinomial Gn (X) = −1 + X + Xn . The sequence of Perron numbers ( θ n − 1 ) n ≥ 2 $(\theta _n^{ - 1} )_{n \ge 2} $ tends to 1. We prove that the Conjecture of Lehmer is true for { θ n − 1 | n ≥ 2 } $\{ \theta _n^{ - 1} |n \ge 2\} $ by the direct method of Poincaré asymptotic expansions (divergent formal series of functions) of the roots θn , zj,n , of Gn (X) lying in |z| < 1, as a function of n, j only. This method, not yet applied to Lehmer’s problem up to the knowledge of the author, is successfully introduced here. It first gives the asymptotic expansion of the Mahler measures M ( G n ) = M ( θ n ) = M ( θ n − 1 ) ${\rm{M}}(G_n ) = {\rm{M}}(\theta _n ) = {\rm{M}}(\theta _n^{ - 1} )$ of the trinomials Gn as a function of n only, without invoking Smyth’s Theorem, and their unique limit point above the smallest Pisot number. Comparison is made with Smyth’s, Boyd’s and Flammang’s previous results. By this method we obtain a direct proof that the conjecture of Schinzel-Zassenhaus is true for { θ n − 1 | n ≥ 2 } $\{ \theta _n^{ - 1} |n \ge 2\} $ , with a minoration of the house , and a minoration of the Mahler measure M(Gn ) better than Dobrowolski’s one. The angular regularity of the roots of Gn , near the unit circle, and limit equidistribution of the conjugates, for n tending to infinity (in the sense of Bilu, Petsche, Pritsker), towards the Haar measure on the unit circle, are described in the context of the Erdős-Turán-Amoroso-Mignotte theory, with uniformly bounded discrepancy functions.