Quantitative correlations and some problems on prime factors of consecutive integers

TL;DR

This paper proves several old conjectures related to prime divisors of consecutive integers, using probabilistic methods, sieve techniques, and recent correlation estimates for multiplicative functions.

Contribution

It establishes the infinitude of integers with bounded prime divisors, proves the irrationality of a series involving $ ext{ω}(n)$, and confirms an asymptotic formula for integers with equal prime divisor counts.

Findings

Infinitely many integers with $ ext{ω}(n+k) ext{ and } ext{Ω}(n+k)$ bounded by $k$

The series $ ext{∑} ext{ω}(n)/2^n$ is irrational

Asymptotic formula for the count of $n extless x$ with $ ext{ω}(n)= ext{ω}(n+1)$

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.