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

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 $a,b:\mathbb{N}\to \mathbb{C}$, $$\frac{1}{\log N}\sum _{n=1}^N\frac{a(\Omega (n))b(\Omega (n+1))}{n}=(\frac{1}{N}\sum _{n=1}^{N}a(\Omega (n)))(\frac{1}{N}\sum _{n=1}^{N}b(\Omega (n)))+{\mathrm{o}}_{N\to \infty }(1).$$ 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 $\ell$.

43 pages, 3rd version (improvement of the error term in the main theorem; appendix added)