AN UNIVERSAL MINORATION OF THE MAHLER MEASURE OF REAL RECIPROCAL ALGEBRAIC INTEGERS

This paper proves the existence of an universal nontrivial minorant of the set of the Mahler measures M(β) where β runs over the set of reciprocal algebraic integers > 1. It represents a part of the solution of the Conjecture of Lehmer, the one restricted to the set of real reciprocal algebraic integers > 1. The proof relies upon, when β tends to one: (i) the asymptotic geometry of the lenticular zeroes of the Parry Upper functions f β (z) associated with the dynamical zeta functions ζ β (z) of the Rényi-Parry arithmetical dynamical systems (β-shift), (ii) the identification of these lenticular zeroes with conjugates of β using Kala-Vavra's Theorem, (iii) a Dobrowolski-type inequality deduced from the Poincaré asymptotic expansions of these lenticular zeroes as a function of the dynamical degree of β. The dynamical zeta function is used to investigate the domain of very small Mahler measures of algebraic integers in the range (1, 1.176280. . .], where 1.176280 is Lehmer's number, if any