Primes in arithmetic progressions with friable indices

Speaker: 吴杰(Université de Lorraine)

Venue: B1032

Time: 2017年12月04日 10:00-11:00

Title: Primes in arithmetic progressions with friable indices

Abstract: We consider the number $pi(x,y;q,a)$ of primes $pleqslant x$ such that $pequivabmod q$ 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.