Title: Note on shifted primes with large prime factors Show Chinese title

Title: 关于具有大素因数的移位素数的注记

Abstract: For any $0<c<1$ let $$ T_c(x)=|\big\{p\le x: p\in \mathbb{P}, P^+(p-1)\ge p^c\big\}|, $$ where $\mathbb{P}$ is the set of primes and $P^+(n)$ denotes the largest prime factor of $n$. Erd\H os proved in 1935 that $$ \limsup_{x\rightarrow \infty}T_c(x)/\pi(x)\rightarrow 0, \quad \text{as~}c\rightarrow 1, $$ where $\pi(x)$ denotes the number of primes not exceeding $x$. Recently, Ding gave a quantitative form of Erd\H os' result and showed that for $8/9< c<1$ we have $$ \limsup_{x\rightarrow \infty}T_c(x)/\pi(x)\le 8\big(c^{-1}-1\big). $$ In this article, Ding's bound is improved to $$ \limsup_{x\rightarrow \infty}T_c(x)/\pi(x)\le -\frac{7}{2}\log c $$ for $e^{-\frac{2}{7}}< c<1$. Show Chinese abstract

Abstract: 对于任何$0<c<1$，令 $$ T_c(x)=|\big\{p\le x: p\in \mathbb{P}, P^+(p-1)\ge p^c\big\}|, $$ 其中$\mathbb{P}$是素数的集合，$P^+(n)$表示$n$的最大素因数。 1935年，Erdős证明了 $$ \limsup_{x\rightarrow \infty}T_c(x)/\pi(x)\rightarrow 0, \quad \text{as~}c\rightarrow 1, $$ 其中 $\pi(x)$表示不超过 $x$的素数个数。 最近，Ding给出了Erdős结果的量化形式，并证明了对于 $8/9< c<1$我们有 $$ \limsup_{x\rightarrow \infty}T_c(x)/\pi(x)\le 8\big(c^{-1}-1\big). $$ 在本文中，Ding的界被改进为 $$ \limsup_{x\rightarrow \infty}T_c(x)/\pi(x)\le -\frac{7}{2}\log c $$ 对于 $e^{-\frac{2}{7}}< c<1$。