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 $\tau(n) = \tau(n+1)$.