Primes Primes in arithmetic progressions with friable indices
DATE:2017-11-30
Speaker
: 吴杰(Université de Lorraine)
Venue
: B1032
Time
: 2017年12月04日 09:00-10:00
Title
: Primes in arithmetic progressions with friable indices
Abstract
: We consider the number $pi(x,y;q,a)$ of primes $pleqslant x$ such that $pequiv abmod q$ and $(p-a)/q$ is free of prime factors larger than $y$. Assume a suitable form of Elliott--Halberstam conjecture, it is proved that $pi(x,y;q,a)$ is asymptotic to $rho(log(x/q)/log y)pi(x)/varphi(q)$ on average, subject to certain ranges of $y$ and $q$, where $rho$ 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.
This is a joint work with Jianya Liu and Ping Xi.
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