The lower bound of shifted primes with large prime factors

Source abstract

In this paper, we consider the asymptotic density of $T_c(x):=\#\{p\leq x:P^+(p-1)\geq p^c\}$, where $P^+(n)$ denote the largest prime factor of $n$. We show that for $x\rightarrow\infty$, for any $0 0$ for $11/32 0.0436$. This improves a previous result by Liu-Wu-Xi (2020), who showed $$\begin{align*} \mathop{\lim\inf}_{x\rightarrow\infty}\frac{T_c(x)}{π(x)}&\geq\max\left(1-4ρ\left(\frac{1}{c}\right),1-c\right), \end{align*}$$ for $0<c\leq 1/2$.