Quantitative correlations and some problems on prime factors of consecutive integers

Abstract

We consider several old problems involving the number of prime divisors function $\omega(n)$, as well as the related functions $\Omega(n)$ and $\tau(n)$. Firstly, we show that there are infinitely many positive integers $n$ such that $\omega(n+k) \leq \Omega(n+k) \ll k$ for all positive integers $k$, establishing a conjecture of Erd\H{o}s and Straus. Secondly, we show that the series $\sum_{n=1}^{\infty} \omega(n)/2^n$ is irrational, settling a conjecture of Erd\H{o}s. Thirdly, we prove an asymptotic formula conjectured by Erd\H{o}s, Pomerance and S\'ark\"ozy for the number of $n\leq x$ satisfying $\omega(n)=\omega(n+1)$, for almost all $x$, with similar results for $\Omega$ and $\tau$. Common to the resolution of all these problems is the use of the probabilistic method. For the first problem, this is combined with computations involving a high-dimensional sieve of Maynard-type. For the second and third problems, we instead make use of a general quantitative estimate for two-point correlations of multiplicative functions with a small power of logarithm saving that may be of independent interest. This correlation estimate is derived by using recent work of Pilatte.