2019/11/24 by Verger-Gaugry, Jean-Louis
#11K16 #11K26 #11K36 #11M41 #11R06 #11R09 #30A12 #30B10 #30B40 #37C30 #58F20 #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1911.10590
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 (β-shift), for α 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 Mr(\houseα) of the Mahler measure \rm M(α), (iii) the Poincaré asymptotic expansions of these poles and of this minorant \rm Mr(\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(α) is obtained. The universal minorant of M(α) obtained is θη-1 > 1, for some integer η≥ 259, where θη is the positive real root of -1+x+xη. 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. For sequences of algebraic integers of Mahler measure smaller than the smallest Pisot number Θ= 1.3247…, 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.