A PROOF OF THE CONJECTURE OF LEHMER

Une démonstration de la Conjecture de Lehmer

Jean-Louis Verger-Gaugry (1)

1 IF - Institut Fourier

The Conjecture of Lehmer is proved to be true. The proof mainly relies upon: (i) the properties of the Parry Upper functions f_{house α}(z) associated with the dynamical zeta functions ζ_{house α} (z) of the Rényi-Parry arithmetical dynamical systems (β-shift), for α a reciprocal algebraic integer α of house {house α} greater than 1, (ii) the discovery of lenticuli of poles of ζ_{house α}(z) which uniformly equidistribute at the limit on a limit "lenticular" arc of the unit circle, when α tends to 1^+ , giving rise to a continuous lenticular minorant M_r(α) of the Mahler measure M(α), (iii) the Poincaré asymptotic expansions of these poles and of this minorant M_r(α) as a function of the dynamical degree. The Conjecture of Schinzel-Zassenhaus is proved to be true. A Dobrowolski type minoration of the Mahler measure M(α) is obtained. The universal minorant of M(α) obtained is (θ_η)^{-1) > 1, for some integer η ≥ 259, where θ_η is the positive real root of −1 + x + x^η. The set of Salem numbers is shown to be bounded from below by the Perron number (θ_{31})^(-1) = 1.08545. . ., dominant root of the trinomial −1 − z^(30) + z^(31). Whether Lehmer's number is the smallest Salem number remains open. For sequences of algebraic integers of Mahler measure smaller than the smallest Pisot number Θ = 1.3247. . ., whose houses have a dynamical degree tending to infinity, the Galois orbit measures of conjugates are proved to converge towards the Haar measure on |z| = 1 (limit equidistribution). The dynamical zeta function is used to investigate the domain of very small Mahler measures of algebraic integers in the range (1, 1.176280], if any

Mots clés: Lehmer conjecture; Schinzel-Zassenhaus conjecture; Mahler measure; mi- noration; Dobrowolski inequality; asymptotic expansion; transfer operator; dynamical zeta function; Rényi-Parry β -shift; Parry Upper function; Perron number; Pisot number; Salem number; Parry number; limit equidistribution

Domaines: Théorie des nombres [math.NT]

Jean-Louis Verger-Gaugry. A PROOF OF THE CONJECTURE OF LEHMER. 2021. ⟨hal-02649184v3⟩