: An attack of the Conjecture of Lehmer by the dynamical zeta function of the β -shift
Résumé
The present work proposes an attack of the Conjecture of Lehmer by the dynamical zeta function of the β -shift to prove that this Conjecture is true (arXiv.org > math > arXiv:1709.03771). In 1933 Lehmer asked the question about the existence of integer polynomials having a Mahler measure different of one, smaller than Lehmer’s number (and arbitrarily close to one). The problem of Lehmer became a Conjecture, stating that there exists a universal lower bound > 1 to the Mahler measures of the nonzero algebraic integers which are not roots of unity. The problem of the minoration of the Mahler measure of algebraic integers is a very deep one and has been extended in the theory of heights in arithmetic geometry.
The main ingredients arise from the R ́enyi-Parry dynamical system of real algebraic numbers and will be biefly presented. They involve: (i) the properties of the Parry Upper functions f_α (z) associated with the dynamical zeta functions ζ_α (z) of the R ́enyi–Parry arithmetical dynamical systems, for α an algebraic integer of house “ α ” greater than 1, tending to one, (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, for α tending to 1 + , giving rise to a continuous lenticular minorant M_r ( α ) of the Mahler measure M(α), (iii) the Poincar ́e asymptotic expansions of these poles and of this minorant M_r ( α ) as a function of the dynamical degree. The same arguments apply to the Conjecture of Schinzel-Zassenhaus, also allow to obtain an inequality improving those of Dobrowolski and Voutier. Whether Lehmer’s number is the smallest Mahler measure > 1 of algebraic integers remains open.
Domaines
Théorie des nombres [math.NT]
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