For any integer n ≥ 2, let P（n） be the largest prime factor of n. In this paper, we prove 1 This that the number of primes p 〈 x with P（p- 1） ≥ pC is more than （1 -c＋o（1））π（x） for 0 〈 c 〈 1/2 extends...For any integer n ≥ 2, let P（n） be the largest prime factor of n. In this paper, we prove 1 This that the number of primes p 〈 x with P（p- 1） ≥ pC is more than （1 -c＋o（1））π（x） for 0 〈 c 〈 1/2 extends a recent result of Luca, Menares and Madariaga for1/4≤c≤1/2. We also pose two conjectures for further research.展开更多
As usual, denote by P(n) the largest prime factor of the integer n 1 with the convention P(1) = 1.For 0 < θ < 1, define Tθ(x) := |{p x : P(p-1) ≥ p~θ}|.In this paper, we obtain a new lower bound for Tθ(x) a...As usual, denote by P(n) the largest prime factor of the integer n 1 with the convention P(1) = 1.For 0 < θ < 1, define Tθ(x) := |{p x : P(p-1) ≥ p~θ}|.In this paper, we obtain a new lower bound for Tθ(x) as x →∞, which improves some recent results of Luca et al.(2015) and Chen and Chen(2017). As a corollary, we partially prove a conjecture of Chen and Chen(2017)about the size of Tθ(x).展开更多
We consider the numberπ(x,y;q,a)of primes p≤such that p≡a(mod q)and(p-a)/q is free of prime factors greater than y.Assuming a suitable form of Elliott-Halberstam conjecture,it is proved thatπ(x,y:q,a)is asymptotic...We consider the numberπ(x,y;q,a)of primes p≤such that p≡a(mod q)and(p-a)/q is free of prime factors greater than y.Assuming a suitable form of Elliott-Halberstam conjecture,it is proved thatπ(x,y:q,a)is asymptotic to p(log(x/q)/log y)π(x)/φ(q)on average,subject to certain ranges of y and q,where p is the Dickman function.Moreover,unconditional upper bounds are also obtained via sieve methods.As a typical application,we may control more effectively the number of shifted primes with large prime factors.展开更多
基金Supported by National Natural Science Foundation of China(Grant Nos.11571174,11401411 and 11371195)
文摘For any integer n ≥ 2, let P（n） be the largest prime factor of n. In this paper, we prove 1 This that the number of primes p 〈 x with P（p- 1） ≥ pC is more than （1 -c＋o（1））π（x） for 0 〈 c 〈 1/2 extends a recent result of Luca, Menares and Madariaga for1/4≤c≤1/2. We also pose two conjectures for further research.
基金supported by Scientific and Technological Research Program of Chongqing Municipal Education Commission(Grant No.KJ1601213)
文摘As usual, denote by P(n) the largest prime factor of the integer n 1 with the convention P(1) = 1.For 0 < θ < 1, define Tθ(x) := |{p x : P(p-1) ≥ p~θ}|.In this paper, we obtain a new lower bound for Tθ(x) as x →∞, which improves some recent results of Luca et al.(2015) and Chen and Chen(2017). As a corollary, we partially prove a conjecture of Chen and Chen(2017)about the size of Tθ(x).
基金supported by the Programme de Recherche Conjoint CNRS-NSFC(Grant No.1457)supported by National Natural Science Foundation of China(Grant No.11531008)+3 种基金the Ministry of Education of China(Grant No.IRT16R43)the Taishan Scholar Project of Shandong Provincesupported by National Natural Science Foundation of China(Grant No.11601413)NSBRP of Shaanxi Province(Grant No.2017JQ1016)
文摘We consider the numberπ(x,y;q,a)of primes p≤such that p≡a(mod q)and(p-a)/q is free of prime factors greater than y.Assuming a suitable form of Elliott-Halberstam conjecture,it is proved thatπ(x,y:q,a)is asymptotic to p(log(x/q)/log y)π(x)/φ(q)on average,subject to certain ranges of y and q,where p is the Dickman function.Moreover,unconditional upper bounds are also obtained via sieve methods.As a typical application,we may control more effectively the number of shifted primes with large prime factors.
