A Proof of the Conjecture of Lehmer and of the Conjecture of Schinzel-Zassenhaus

2017/09/12 by Jean-Louis Verger-Gaugry, Verger-Gaugry, Jean-Louis · 1 voice · 1 citation
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #Quantum chaos and dynamical systems #math.NT

paper · pdf · doi:10.48550/arxiv.1709.03771

openalex publication_date 2017/09/12 · arxiv published 2017/09/12 · arxiv updated 2017/09/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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, for α an 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 house(α) tends to 1⁺, giving rise to a continuous lenticular minorant \rm M_r(house(α)) of the Mahler measure \rm M(α), (iii) the Poincaré asymptotic expansions of these poles and of this minorant \rm M_r(house(α)) as a function of the dynamical degree. With the same arguments the conjecture of Schinzel-Zassenhaus is proved to be true. An inequality improving those of Dobrowolski and Voutier ones is obtained. The set of Salem numbers is shown to be bounded from below by the Perron number θ₃₁^-1 = 1.08545…, dominant root of the trinomial -1 - z³⁰ + z³¹. Whether Lehmer's number is the smallest Salem number remains open. A lower bound for the Weil height of nonzero totally real algebraic numbers, ≠ ± 1, is obtained (Bogomolov property). For sequences of algebraic integers of Mahler measure smaller than the smallest Pisot number, 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).

Citations

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Cited by

• A remark on a recent proof of Lehmer's Conjecture [arXiv:1709.03771]

Discussions

• A (claimed) Proof of Lehmer's Conjecture [hn, 3 points, 0 comments]

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