On the least almost-prime in arithmetic progressions. (English) Zbl 07893397
Summary: Let \(\mathcal{P}_{2}\) denote a positive integer with at most \(2\) prime factors, counted according to multiplicity. For integers \(a\), \(q\) such that \((a,q)=1\), let \(\mathcal{P}_{2}(q,a)\) denote the least \(\mathcal{P}_{2}\) in the arithmetic progression \(\{nq+a\}_{n=1}^{\infty}\). It is proved that for sufficiently large \(q\), we have \[\mathcal{P}_{2}(q,a)\ll q^{1.825}.\] This result constitutes an improvement upon that of J. Li et al. [Czech. Math. J. 73, No. 1, 177–188 (2023; Zbl 07655761)], who obtained \(\mathcal P_{2}(q,a)\ll q^{1.8345}\).
Citations:
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