# Primes in Arithmetic Progressions with Friable Indices and Applications

In this talk, we shall present our recent works on primes in arithmetic progressions with friable indices, joint with Jianya Liu and Ping Xi. Denote by $\pi(x,y;q,a)$ the number of primes $p\leqslant x$ such that $p\equiv a\bmod q$ 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 applications, we shall consider the following two problems :
1) the number of shifted primes with large prime factors,
2) friable variant of the Titchmarsh divisor problem.

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