An Attack of the Conjecture of Lehmer by the Dynamical Zeta function of the β -shift

selected oral communicationInternational audienceThe present work proposes an attack of the Conjecture of Lehmer by the dynamical zeta function of the beta-shift to prove that this Conjecture is true (now: https://hal.archives-ouvertes.fr/hal-03119536v1). 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 Rényi-Parry dynamical system of real algebraic numbers and will be briefly presented. They involve: (i) the properties of the Parry Upper functions f_house(alpha) (z) associated with the dynamical zeta functions zeta_house(alpha) (z) of the Rényi-Parry arithmetical dynamical systems, for alpha an algebraic integer of house “ house(a) ” greater than 1, tending to one, (ii) the discovery of lenticuli of poles of zeta_house(alpha) (z) which uniformly equidistribute at the limit on a limit “lenticular” arc of the unit circle, for house(alpha) tending to 1+, giving rise to a continuous lenticular minorant Mr( house(a) ) of the Mahler measure M(alpha), (iii) the Poincaré asymptotic expansions of these poles and of this minorant Mr(house(alpha) ) 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