The lower bound of shifted primes with large prime factors

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 $0<c\leq 1/2$, one has \begin{align*} \mathop{\lim\inf}_{x\rightarrow\infty}\frac{T_c(x)}{\pi(x)}&\geq\max\left(1-\frac{16}{5}\rho\left(\frac{1}{c}\right),1-c+g(c)\right), \end{align*} where $g(c)>0$ for $11/32<c\leq1/2$, and $g(c)=0$ for $0<c\leq11/32$. In particular, we have $g(1/2)>0.0436$. This improves a previous result by Liu-Wu-Xi (2020), who showed \begin{align*} \mathop{\lim\inf}_{x\rightarrow\infty}\frac{T_c(x)}{\pi(x)}&\geq\max\left(1-4\rho\left(\frac{1}{c}\right),1-c\right), \end{align*} for $0<c\leq 1/2$.