Speaker: 吴杰（CNRS）

Venue: B1044

Time: 2018年5月29日 10:00-11:00

Title: On values taken by the largest prime factor of shifted primes

Abstract: Denote by $\P$ the set of all prime numbers and by $P(n)$ the largest prime factor of positive integer $n\ge 1$ with the convention $P(1)=1$. In this talk, we shall consider the distribution of the largest prime factor of shifted primes and present some results on this topics. In particular, we shall prove that for each $\eta\in (\tfrac{32}{17}, \; 2.1426\dots)$, there is a constant $c(\eta)>1$ such that for every fixed non-zero integer $a\in \Z^*$ the set $$\{p\in \P \,:\, p=P(q-a) \; \text{for some prime $q$ with $p^{\eta}<q\le c(\eta)p^{\eta}$}\} $$ has relative asymptotic density 1 in $\P$. This improves a similar result due to Banks \& Shparlinski (2007), which requires $\eta\in (\tfrac{32}{17}, \; 2.0606\dots)$ in place of $\eta\in (\tfrac{32}{17}, \; 2.1426\dots)$.