Let $n ≥ 2$ 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 $(\theta_{n}^{−1} )_{n≥2}$ tends to 1. We prove thatthe Conjecture of Lehmer is true for $\{\theta_{n}^{−1} | n ≥ 2\}$ by the direct method of Poincar\'e asymptoticexpansions (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 $n, j$ only. This method, not yet applied to Lehmer’s problem up to theknowledge of the author, is successfully introduced here. It first gives the asymptotic expansionof the Mahler measures ${\rm M}(G_n) = {\rm M}(\theta_{n}) = {\rm M}(\theta_{n}^{-1})$ of the trinomials $G_n$ as a function of $n$only, without invoking Smyth’s Theorem, and their unique limit point above the smallest Pisotnumber. Comparison is made with Smyth’s, Boyd’s and Flammang’s previous results. Bythis method we obtain a direct proof that the conjecture of Schinzel-Zassenhaus is true for$\{\theta_{n}^{−1} | n ≥ 2\}$, with a minoration of the house $\house\{\theta_{n}^{−1}\}= \theta_{n}^{−1}$ , and a minoration of the Mahler measure${\rm M}(G_n)$ better than Dobrowolski’s one for $\{\theta_{n}^{−1} | n ≥ 2\}$ . The angular regularity of the roots of $G_n$ , near the unitcircle, 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 ofthe Erd\H{o}s-Tur\'an-Amoroso-Mignotte theory, with uniformly bounded discrepancy functions.