Title: A Proof of the Conjecture of Lehmer and of the Conjecture of Schinzel-Zassenhaus,
Abstract: The conjecture of Lehmer is proved to be true. The proof mainly relies upon:
(i) the properties of the Parry Upper functions $f_{house(\alpha)}(z)$
associated with the dynamical zeta functions $\zeta_{house(\alpha)}(z)$ of the
Rényi--Parry arithmetical dynamical systems, for $\alpha$ an algebraic
integer $\alpha$ of house "$house(\alpha)$" greater than 1, (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,
when $house(\alpha)$ tends to $1^+$, giving rise to a continuous lenticular
minorant ${\rm M}_{r}(house(\alpha))$ of the Mahler measure ${\rm M}(\alpha)$,
(iii) the Poincaré asymptotic expansions of these poles and of this minorant
${\rm M}_{r}(house(\alpha))$ 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
$\theta_{31}^{-1} = 1.08545\ldots$, dominant root of the trinomial $-1 - z^{30}
+ z^{31}$. Whether Lehmer's number is the smallest Salem number remains open. A
lower bound for the Weil height of nonzero totally real algebraic numbers,
$\neq \pm 1$, 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).