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
This special issue on Analytic Number Theory is dedicated to Jing-run Chen's Theorem(1+2)on the Goldbach Conjecture,the proof of which was first published in SCIENCE CHINA Mathematics exactly 50 years ago.Jing-run...This special issue on Analytic Number Theory is dedicated to Jing-run Chen's Theorem(1+2)on the Goldbach Conjecture,the proof of which was first published in SCIENCE CHINA Mathematics exactly 50 years ago.Jing-run Chen was born on May 22,1933 in Fujian province,China.In 1953,he graduated from Xiamen University with a B.Sc.degree in Mathematics.展开更多
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.
文摘This special issue on Analytic Number Theory is dedicated to Jing-run Chen's Theorem(1+2)on the Goldbach Conjecture,the proof of which was first published in SCIENCE CHINA Mathematics exactly 50 years ago.Jing-run Chen was born on May 22,1933 in Fujian province,China.In 1953,he graduated from Xiamen University with a B.Sc.degree in Mathematics.
