"The 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 (math NT> arXiv:1911.10590(29 Oct 2021)). In 1933 Lehmer asked the question about the existence of integer polynomials having a Mahler measure different of one, smaller than Lehmers 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 lenticular poles of the dynamical zeta functions $\zeta_\beta(z)$ of the RényiParry arithmetical dynamical ($\beta$-shift), with $\beta> 1$ any real number tending to one, to which a lenticular measure can be associated, satisfying a Dobrowolski-type inequality with the dynamical degree of $\beta$ . When $\beta$ runs over the set of nonzero reciprocal algebraic integers, under some assumptions, the lenticular poles are identified with conjugates of $\beta$, using Kala-Vavras periodic representation theorem (2019), and this lenticular measure is identified with a minorant of the Mahler measure of $\beta$. Though expressed as hypergeometric functions (Mellin, 1915) the lenticularity of the poles only appears when using their Poincaré asymptotic expansions, in the angular sector guessed by M. Langevin, G. Rhin and C. Smyth, G. Rhin and Q. Wu. We show that the search for very small Mahler measures calls for investigating the factorization of integer polynomials in a class of lacunary polynomials canonically associated to the functions $\zeta_\beta(z)$, that this problem is linked to the number of zeroes of these polynomial in $\mathbb{F}_p$, to their asymptotic limit when $p$ tends to infinity, and questions on the existence of modular forms by the Langlands program. Whether Lehmers number is the smallest Mahler measure $>1$ of algebraic integers remains open."