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

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 Erdős and Turán’s conjectures asserts that the asymptotic density of integers $n$ satisfying $P^{+}(n)<P^{+}(n+1)$ is $1\mathrm{/}2$. Furthermore, Wang conjectures that for $x\to \mathrm{\infty }$, one has $\mathrm{\#}\{p\le x:P^{+}(p-1)<P^{+}(p+1)\}\sim \pi (x)\mathrm{/}2$. In this paper, we show the following results. For any $3\le y\ll x^{o(1)}$, for $x\to \mathrm{\infty }$, 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}π(x). \end{align*} For any $0<\alpha <17\mathrm{/}32$, 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(α)+o(1))π(x). \end{align*} In particular, the function $h$ satisfies ${\lim }_{\alpha \to 0^{+}}h(\alpha )=1\mathrm{/}2$. Similar result also holds for $\mathrm{\#}\{n\le x:P_y^{+}(n)<P_y^{+}(n+1)\}$. These improve Rivat’s result (2001) and Wang’s result (2019).