**1. Introduction**

In [1,2], we constructed-and-studied *weighted-semicircular elements* and *semicircular elements* induced by *p*-*adic number fields* Q*<sup>p</sup>*, for all *p* ∈ P, where P is the set of all *primes* in the set N of all *natural numbers*. In this paper, we consider certain "truncated" free-probabilistic information of the weighted-semicircular laws and the semicircular law of [1]. In particular, we are interested in free distributions of certain free reduced words in our (weighted-)semicircular elements under conditions dictated by the primes *p* in a "suitable" *closed interval* [*<sup>t</sup>*1, *t*2] of the set R of *real numbers*. Our results illustrate how the original (weighted-)semicircular law(s) of [1] is (resp., are) distorted by truncations on P.

### *1.1. Preview and Motivation*

Relations between *primes* and *operators* have been widely studied not only in mathematical fields (e.g., [3–6]), but also in other scientific fields (e.g., [7]). For instance, we studied how primes act on certain *von Neumann algebras* generated by *p* -adic and Adelic *measure spaces* in [8,9]. Meanwhile, in [10], primes are regarded as *linear functionals* acting on *arithmetic functions*, understood as *Krein-space operators* under the representation of [11]. Furthermore, in [12,13], free-probabilistic structures on *Hecke algebras* H -*GL*2(Q*p*) are studied for *p* ∈ P. These series of works are motivated by number-theoretic results (e.g., [4,5,7]).

In [2], we constructed weighted-semicircular elements {*Qp*,*j*}*j*∈<sup>Z</sup> and corresponding semicircular elements {<sup>Θ</sup>*p*,*j*}*j*∈<sup>Z</sup> in a certain Banach ∗-algebra LS*p* induced from the <sup>∗</sup>-*algebra* M*p* consisting of *measurable functions* on a *p*-adic number field Q*<sup>p</sup>*, for *p* ∈ P. In [1], the *free product* Banach ∗-probability space -LS, *τ*0 of the measure spaces {LS*p*(*j*)}*p*∈P,*j*∈<sup>Z</sup> of [2] were constructed over both primes and integers, and weighted-semicircular elements {*Qp*,*j*}*p*∈P, *j*∈Z and semicircular elements {<sup>Θ</sup>*p*,*j*}*p*∈P, *j*∈Z were studied in LS, as *free generators.*

In this paper, we are interested in the cases where the free product linear functional *τ*0 of [1] on the Banach ∗-algebra LS is truncated in P. The distorted free-distributional data from such truncations are considered. The main results characterize how the original free distributions on -LS, *τ*0 are affected by the given truncations on P.
