Asymptotic independence of $Ω(n)$ and $Ω(n+1)$ along logarithmic averages
Dimitrios Charamaras, Florian K. Richter
TL;DR
The paper proves that, for bounded functions $a,b$, the logarithmically averaged joint distribution of $\Omega(n)$ and $\Omega(n+1)$ factors asymptotically into the product of their individual logarithmic averages, providing a quantitative bound with double-logarithmic savings. The authors reduce the problem to averages involving completely multiplicative functions $f_{\xi,N}$, handle small and large frequency contributions separately via a local Erdős–Kac bound and Halász/MRT15 techniques, and combine circle method arguments to control error terms. This yields a natural form of asymptotic independence between consecutive values of $\Omega(n)$ and has applications to the distribution of $\Omega(p+1)$ for almost primes. The work also frames conjectures about independence for higher-order prime-factor statistics and establishes equivalence relations with functional Chowla-type statements. Overall, the paper extends Tao’s logarithmic two-point correlation results to the $\Omega(n)$-level setting with explicit, double-logarithmic savings, advancing understanding of independence phenomena in multiplicative-integer factorization.
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 $\ell$.