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On the least almost-prime in arithmetic progression. (English) Zbl 07655761

Summary: Let \(\mathcal{P}_r\) denote an almost-prime with at most \(r\) prime factors, counted according to multiplicity. Suppose that \(a\) and \(q\) are positive integers satisfying \((a,q)=1\). Denote by \(\mathcal{P}_2(a,q)\) the least almost-prime \(\mathcal{P}_2\) which satisfies \(\mathcal{P}_2\equiv a\pmod q\). It is proved that for sufficiently large \(q\), there holds \[\mathcal{P}_2(a,q)\ll q^{1.8345}.\] This result constitutes an improvement upon that of H. Iwaniec [J. Math. Soc. Japan 34, 95–123 (1982; Zbl 0486.10033)], who obtained the same conclusion, but for the range \(1.845\) in place of \(1.8345\).

MSC:

11N13 Primes in congruence classes
11N35 Sieves
11N36 Applications of sieve methods

Citations:

Zbl 0486.10033
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References:

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