Integers, without large prime factors, in arithmetic progressions, I

We show that, for any fixed ε > 0, there are asymptotically the same number of integers up to x, that are composed only of primes ⩽ y, in each arithmetic progression (mod q), provided that y ⩾ q1+e and log x/log q →∞ as y→∞: this improves on previous estimates.

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