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 $f_{\houseα}(z)$ associated with the dynamical zeta functions $ζ_{\houseα}(z)$ of the Rényi--Parry arithmetical dynamical systems ($\beta$-shift), for $\alpha$ a reciprocal algebraic integer of house $\houseα$ greater than 1, (ii) the discovery of lenticuli of poles of $ζ_{\houseα}(z)$ which uniformly equidistribute at the limit on a limit "lenticular" arc of the unit circle, when $\houseα$ tends to $1^{+}$, giving rise to a continuous lenticular minorant ${\rm M}_{r}(\houseα)$ of the Mahler measure $\mathrm{M}(\alpha )$, (iii) the Poincaré asymptotic expansions of these poles and of this minorant ${\rm M}_{r}(\houseα)$ 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$(\alpha )$ is obtained. The universal minorant of M$(\alpha )$ obtained is ${\theta }_{\eta }^{-1}>1$, for some integer $\eta \ge 259$, where ${\theta }_{\eta }$ is the positive real root of $-1+x+x^{\eta }$. The set of Salem numbers is shown to be bounded from below by the Perron number ${\theta }_{31}^{-1}=1.08545\dots$, 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 $\Theta =1.3247\dots$, 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.

Results unchanged, revised arguments in Section 5. "Mahler measures M(beta) < 1.176280" indicated explicitely everywhere. Theorem 10.1 and its proof: revised. arXiv admin note: substantial text overlap with arXiv:1709.03771

References in corpus (3)

• Survey article: Seventy years of Salem numbers
• Uniform distribution of Galois conjugates and beta-conjugates of a Parry number near the unit circle and dichotomy of Perron numbers
• Beta-conjugates of real algebraic numbers as Puiseux expansions

Cited by in corpus (1)

• Errata and Addenda to Mathematical Constants