A proof of the Conjecture of Lehmer

Abstract

The Conjecture of Lehmer is proved to be true. The proof mainly relies upon: (i) the properties of the Parry Upper functions f α (z) associated with the dynamical zeta functions ζ α (z) of the Rényi-Parry arithmetical dynamical systems (β -shift), for α a reciprocal algebraic integer α of house α greater than 1, (ii) the discovery of lenticuli of poles of ζ 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 −1 η > 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 −1 31 = 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).