On the Conjecture of Lehmer, Limit Mahler Measure of Trinomials and Asymptotic Expansions
Abstract Let n 2 be an integer and denote by thn the real root in (0, 1) of the trinomial Gn(X) 1 X Xn. The sequence of Perron numbers (thn 1)n>=2 \theta _n^ 1} )_{n \ge 2} tends to 1. We prove that the Conjecture of Lehmer is true for {thn 1|n>=2} \theta _n^ 1} |n \ge 2\ by the direct method of Poincare asymptotic expansions (divergent formal series of functions) of the roots thn, 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(Gn)=M(thn)=M(thn 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 {thn 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 Erdos Turan Amoroso Mignotte theory, with uniformly bounded discrepancy functions.