Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the Bivariate Erd\H{o}s-Kac Theorem and Correlations of the M\"obius
  Function   Let $2 \leq y \leq x$ such that $\beta := \frac{\log x}{\log y} \rightarrow
\infty$. Let $\omega_y(n)$ denote the number of distinct prime factors $p$ of
$n$ such that $p \leq y$, and let $\mu_y(n) := \mu^2(n)(-1)^{\omega_y(n)}$,
where $\mu$ is the M\"{o}bius function. We prove that if $\beta$ is not too
large (in terms of $x$) then for each fixed $a \in \mathbb{N}$,
\begin{equation*} \sum_{n \leq x} \mu_y(n)\mu_y(n+a) \ll x\left(\frac{1}{\log_2
y} + e^{-\frac{1}{21}\beta \log \beta}\right). \end{equation*} This can be seen
as a partial result towards the binary Chowla conjecture. Our main input is a
\emph{quantitative} bivariate analogue of the Erd\H{o}s-Kac theorem regarding
the distribution of the pairs $(\omega(n),\omega(n+a))$, where $n$ and $n+a$
both belong to any subset of the positive integers with suitable sieving
properties; moreover, we show that the set of squarefree integers is an example
of such a set. We end with a further application of this probabilistic result
related to a problem of Erd\H{o}s and Mirsky on the number of integers $n \leq
x$ such that $\tau(n) = \tau(n+1)$.

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The categories of this paper are: chem-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: eess
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The categories of this paper are: cs
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The categories of this paper are: hep-ex
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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: plasm-ph
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The categories of this paper are: physics
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The categories of this paper are: bayes-an
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The categories of this paper are: cmp-lg
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The categories of this paper are: cond-mat
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The categories of this paper are: adap-org
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The categories of this paper are: nlin
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The categories of this paper are: funct-an
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The categories of this paper are: stat