Primes in arithmetic progressions with friable indices

Abstract

We consider the number $\pi(x,y;q,a)$ of primes $p\leqslant~x$such that $p\equiv~a~({\rm~mod}q)$ and $(p-a)/q$ is free of prime factors greater than $y$.Assuming a suitable form of Elliott-Halberstam conjecture,it is proved 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.