题目：Primes in Arithmetic Progressions with Friable Indices and Applications
报告人：吴杰 教授 （CNRS, Université Paris-Est Créteil）
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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