  The reduction modulo $p$ of a family of lacunary integer polynomials,
associated with the dynamical zeta function $\zeta_{\beta}(z)$ of the
$\beta$-shift, for $\beta > 1$ close to one, is investigated. We briefly recall
how this family is correlated to the problem of Lehmer. A variety of questions
is raised about their numbers of zeroes in $\mathbb{F}_p$ and their
factorizations, via Kronecker's Average Value Theorem (viewed as an analog of
classical Theorems of Uniform Distribution Theory). These questions are
partially answered using results of Schinzel, revisited by Sawin, Shusterman
and Stoll, and density theorems (Frobenius, Chebotarev, Serre, Rosen). These
questions arise from the search for the existence of integer polynomials of
Mahler measure > 1 less than the smallest Salem number 1.176280. Explicit
connection with modular forms (or modular representations) of the numbers of
zeroes of these polynomials in $\mathbb{F}_p$ is obtained in a few cases. In
general it is expected since it must exist according to the Langlands program.
