On the largest prime factors less than $y$ of consecutive shifted primes

Abstract

For an integer $n>1$, let $P^+(n)$ be the largest prime factor of $n$, and let $P_y^+(n)$ denote the largest prime factor of $n$ not exceeding $y$. One of Erd\H{o}s and Tur\'an's conjectures asserts that the asymptotic density of integers $n$ satisfying $P^+(n)<P^+(n+1)$ is $1/2$. Furthermore, Wang conjectures that for $x\rightarrow\infty$, one has $\#\{p\leq x:P^+(p-1)<P^+(p+1)\}\sim \pi(x)/2$. In this paper, we show the following results. For any $3\leq y\ll x^{o(1)}$, for $x\rightarrow\infty$, we have \begin{align*}&\#\{n\leq x:P^+_y(n)<P^+_y(n+1)\}\sim \frac{1}{2}x, \\&\#\{p\leq x:P^+_y(p-1)<P^+_y(p+1)\}\sim \frac{1}{2}\pi(x). \end{align*} For any $0<\alpha<17/32$, let $y=x^\alpha$. Then there exists $h(\alpha)>0$ such that \begin{align*} \#\{p\leq x:P_y^+(p-1)<P_y^+(p+1)\}\geq(h(\alpha)+o(1))\pi(x). \end{align*} In particular, the function $h$ satisfies $\lim_{\alpha\rightarrow 0^+}h(\alpha)=1/2$. Similar result also holds for $\#\{n\leq x:P_y^+(n)<P_y^+(n+1)\}$. These improve Rivat's result (2001) and Wang's result (2019).