On the Bivariate Erdős-Kac Theorem and Correlations of the Möbius Function

Abstract

Let $2\le y\le x$ such that $\beta :=\frac{\log x}{\log y}\to \infty$. Let ${\omega }_y(n)$ denote the number of distinct prime factors $p$ of $n$ such that $p\le y$, and let ${\mu }_y(n):={\mu }^2(n)(-1{)}^{{\omega }_y(n)}$, where $\mu$ is the Möbius 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} μ_y(n)μ_y(n+a) \ll x\left(\frac{1}{\log_2 y} + e^{-\frac{1}{21}β\log β}\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 Erdős-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 Erdős and Mirsky on the number of integers $n\le x$ such that $\tau (n)=\tau (n+1)$.

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