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

Abstract

Let $Ω(n)$ denote the number of prime factors of a positive integer $n$ counted with multiplicities. We show that for any bounded functions $a,b\colon\mathbb{N}\to\mathbb{C}$, $$\frac{1}{\log{N}}\sum_{n=1}^N \frac{a(Ω(n))b(Ω(n+1))}{n} = \Bigg(\frac{1}{N}\sum_{n=1}^N a(Ω(n))\Bigg)\Bigg(\frac{1}{N}\sum_{n=1}^N b(Ω(n))\Bigg) + \mathrm{o}_{N\to\infty}(1).$$ 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 $Ω(p+1)$ as $p$ ranges over $\ell$-almost primes for a "typical" value of $\ell$.