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{\Large Department of Mathematics, BGU}

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{\Huge BGU Probability and Ergodic Theory  (PET) seminar}\\[0.2\baselineskip]

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\textbf{On} \emph{Thursday, December 20, 2018}
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\textbf{At} \emph{11:00 -- 12:00}
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\textbf{In} \emph{-101}

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{\large\scshape Ross Pinsky 
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will talk about
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{\Large\bfseries A Natural probabilistic model on the integers and its relation to  Dickman-type distributions and Buchstab's function\par}
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\textsc{Abstract:}
Let $\{p_j\}_{j=1}^\infty$ denote the set of prime numbers in increasing order, let $\Omega_N\subset \mathbb{N}$ denote the set of positive integers with no prime factor larger than $p_N$ and
let $P_N$ denote the probability  measure  on $\Omega_N$ which gives to each $n\in\Omega_N$ a probability proportional to $\frac{1}{n}$.
This measure is in fact the distribution of the random integer $I_N\in\Omega_N$ defined by $I_N=\prod_{j=1}^Np_j^{X_{p_j}}$, where
$\{X_{p_j}\}_{j=1}^\infty$ are independent random variables and $X_{p_j}$ is distributed as Geom$(1-\frac{1}{p_j})$.
We show that $\frac{\log n}{\log N}$ under $P_N$ converges weakly to the \emph{Dickman distribution}. As a corollary, we recover a classical result from classical multiplicative number theory---\emph{Mertens'
formula}, which states that $\sum_{n\in\Omega_N}\frac{1}{n}\sim e^\gamma\log N$ as $N\to\infty$.

Let \$D\_\{\textbackslash{}text\{nat\}\}(A)\$ denote the natural density of \$A\textbackslash{}subset\textbackslash{}mathbb\{N\}\$, if it exists, and let $D_{\text{log-indep}}(A)=\lim_{N\to\infty}P_N(A\cap\Omega_N)$ denote the
density of \$A\$ arising from $\{P_N\}_{N=1}^\infty$, if it exists. We show that the two densities coincide on a natural algebra of subsets of \$\textbackslash{}mathbb\{N\}\$.
We also show that they do not agree on the sets of $n^\frac{1}{s}$- \emph{smooth numbers} $\{n\in\mathbb{N}: p^+(n)\le n^\frac{1}{s}\}$, \$s\textgreater{}1\$, where $p^+(n)$ is the largest prime divisor of \$n\$.
This last  consideration concerns  distributions involving the \emph{Dickman function}.
We also consider the
sets of  \$n\^{}\textbackslash{}frac\{1\}\{s\}\$- \emph{rough numbers}  \$\{n\textbackslash{}in\textbackslash{}mathbb\{N\}:p\^{}-(n)\textbackslash{}ge n\^{}\{\textbackslash{}frac\{1\}\{s\}\}\}\$, \$s\textgreater{}1\$,  where \$p\^{}-(n)\$ is the smallest prime divisor of \$n\$.
We show that the probabilities of these sets, under
the uniform distribution on \${[}N{]}=\{1,\textbackslash{}ldots, N\}\$ and under the \$P\_N\$-distribution on \$\textbackslash{}Omega\_N\$, have the  same
asymptotic decay profile as functions of \$s\$, although their rates are necessarily different. This profile involves  the \emph{Buchstab function}. We also prove a new representation for the Buchstab function.








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