On the Bivariate Erd\H{o}s-Kac Theorem and Correlations of the M\"obius Function

TL;DR

This paper proves a partial result related to the binary Chowla conjecture by analyzing correlations of the Möbius function using a bivariate Erdős-Kac theorem for integers with certain sieving properties.

Contribution

It introduces a quantitative bivariate Erdős-Kac theorem for pairs of integers with sieving properties, advancing understanding of Möbius function correlations.

Findings

Bound on the sum of Möbius function correlations for shifted integers

Demonstration that squarefree integers form a suitable set for the theorem

Application to a problem on the equality of divisor functions for consecutive integers

Abstract

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