Let \( n \geq 2 \) be an integer and denote by \( \theta_n \) the real root in \( (0,1) \) of the trinomial \( G_n(X) = -1 + X + X^n \). The sequence of Perron numbers \( (\theta_n^{-1})_{n \geq 2} \) tends to 1. We prove that the Conjecture of Lehmer is true for \( \{\theta_n^{-1} \mid n \geq 2\} \) by the direct method of Poincaré asymptotic expansions (divergent formal series of functions) of the roots \( \theta_n, z_{j,n} \), of \( G_n(X) \) lying in \( |z| < 1 \), as a function of \( n, j \). 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 \( M(G_n) = M(\theta_n) = M(\theta_n^{-1}) \) of the trinomials \( G_n \) 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 \( \{\theta_n^{-1} \mid n \geq 2\} \), with a minoration of the house \( \overline{|\theta_n^{-1}|} \), and a minoration of the Mahler measure \( M(G_n) \) better than Dobrowolski's one. The angular regularity of the roots of \( G_n \), 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.