Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Syracuse Random Variables and the Periodic Points of Collatz-type maps   Let $p$ be an odd prime, and consider the map $H_{p}$ which sends an integer
$x$ to either $\frac{x}{2}$ or $\frac{px+1}{2}$ depending on whether $x$ is
even or odd. The values at $x=0$ of arbitrary composition sequences of the maps
$\frac{x}{2}$ and $\frac{px+1}{2}$ can be parameterized over the $2$-adic
integers ($\mathbb{Z}_{2}$) leading to a continuous function from
$\mathbb{Z}_2$ to $\mathbb{Z}_p$ which the author calls the numen of $H_{p}$,
denoted $\chi_p$; the $p=3$ case turns out to be an alternative version of the
Syracuse Random Variables constructed by Tao [arXiv:1909.03562]. This paper
establishes the Correspondence Theorem, which shows that an odd integer
$\omega$ is a periodic point of $H_p$ if and only if
$\omega=\chi_{p}\left(n\right)/\left(1-r_{p}\left(n\right)\right)$ for some
integer $n\geq1$, where $r_{p}\left(n\right) = p^{\#_{1}\left(n\right)} /
2^{\lambda\left(n\right)}$, where $\#_1 \left(n\right)$ is the number of $1$s
digits in the binary expansion of $n$ and $\lambda\left(n\right)$ is the number
of digits in the binary expansion of $n$. Using this fact, the bulk of the
paper is devoted to examining the Dirichlet series associated to $\chi_p$ and
$r_p$, which are used along with Perron's Formula to reformulate the
Correspondence Theorem in terms of contour integrals, to which Residue calculus
is applied so as to obtain asymptotic formulae for the quantities therein. A
sampling of other minor results on $\chi_p$ are also discussed in the paper's
final section.

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The categories of this paper are: patt-sol
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The categories of this paper are: nucl-th
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The categories of this paper are: comp-gas
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The categories of this paper are: ao-sci
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The categories of this paper are: mtrl-th
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The categories of this paper are: adap-org
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The categories of this paper are: hep-lat
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The categories of this paper are: plasm-ph
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The categories of this paper are: solv-int
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The categories of this paper are: physics
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The categories of this paper are: math-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: acc-phys
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The categories of this paper are: astro-ph
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The categories of this paper are: q-fin