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

Source 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 Erdős and Turán's conjectures asserts that the asymptotic density of integers $n$ satisfying $P^+(n) 0$ 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 $\lim_{α\rightarrow 0^+}h(α)=1/2$. 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).