  Concentration and equi-distribution, near the unit circle, in Solomyak's set,
of the union of the Galois conjugates and the beta-conjugates of a Parry number
$\beta$ are characterized by means of the Erd\H{o}s-Tur\'an approach, and its
improvements by Mignotte and Amoroso, applied to the analytical function
$f_{\beta}(z) = -1 + \sum_{i \geq 1} t_i z^i$ associated with the R\'enyi
$\beta$-expansion $d_{\beta}(1)= 0.t_1 t_2 ...$ of unity. Mignotte's
discrepancy function requires the knowledge of the factorization of the Parry
polynomial of $\beta$. This one is investigated using theorems of Cassels,
Dobrowolski, Pinner and Vaaler, Smyth, Schinzel in terms of cyclotomic,
reciprocal non-cyclotomic and non-reciprocal factors. An upper bound of
Mignotte's discrepancy function which arises from the beta-conjugates of
$\beta$ which are roots of cyclotomic factors is linked to the Riemann
hypothesis, following Amoroso. An equidistribution limit theorem, following
Bilu's theorem, is formulated for the concentration phenomenon of conjugates of
Parry numbers near the unit circle. Parry numbers are Perron numbers. Open
problems on non-Parry Perron numbers are mentioned in the context of the
existence of non-unique factorizations of elements of number fields into
irreducible Perron numbers (Lind).
