Primes in arithmetic progressions with friable indices

We consider the number π(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.