Quantitative correlations and some problems on prime factors of consecutive integers

Abstract

We consider several old problems involving the number of prime divisors function $\ddot{I}‰(n)$, as well as the related functions $\hat{I}\text{c◯}(n)$ and $\ddot{I}„(n)$. Firstly, we show that there are infinitely many positive integers $n$ such that $\ddot{I}‰(n+k)\le \hat{I}\text{c◯}(n+k)\ll k$ for all positive integers $k$, establishing a conjecture of Erdős and Straus. Secondly, we show that the series ${\sum }_{n=1}^{\infty }\ddot{I}‰(n)/2^n$ is irrational, settling a conjecture of Erdős. Thirdly, we prove an asymptotic formula conjectured by Erdős, Pomerance and Sárközy for the number of $n\le x$ satisfying $\ddot{I}‰(n)=\ddot{I}‰(n+1)$, for almost all $x$, with similar results for $\hat{I}\text{c◯}$ and $\ddot{I}„$. 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.

61 pages; minor edits