Asymptotic independence of $\Omega(n)$ and $\Omega(n+1)$ along logarithmic averages

TL;DR

This paper proves that the number of prime factors of consecutive integers are asymptotically independent along logarithmic averages, extending Tao's work on the Chowla conjecture and providing quantitative error bounds.

Contribution

It generalizes Tao's theorem on the logarithmic correlation of the number of prime factors, establishing asymptotic independence with explicit error terms.

Findings

Asymptotic independence of (n) and (n+1) along logarithmic averages

Quantitative bounds with double-logarithmic savings

New results on the distribution of (p+1) for -almost primes

Abstract

Let $\Omega(n)$ denote the number of prime factors of a positive integer $n$ counted with multiplicities. We show that for any bounded functions $a,b\colon\mathbb{N}\to\mathbb{C}$, $$\frac{1}{\log{N}}\sum_{n=1}^N \frac{a(\Omega(n))b(\Omega(n+1))}{n} = \Bigg(\frac{1}{N}\sum_{n=1}^N a(\Omega(n))\Bigg)\Bigg(\frac{1}{N}\sum_{n=1}^N b(\Omega(n))\Bigg) + \mathrm{o}_{N\to\infty}(1).$$ This generalizes a theorem of Tao on the logarithmically averaged two-point correlation Chowla conjecture. Our result is quantitative and the explicit error term that we obtain establishes double-logarithmic savings. As an application, we obtain new results about the distribution of $\Omega(p+1)$ as $p$ ranges over $\ell$-almost primes for a "typical" value of $\ell$.