Almost-prime values of reducible polynomials at prime arguments (Q2291710)

This paper studies almost-prime values produced by products of irreducible polynomials evaluated at prime arguments. The following theorem is established. Theorem. Let \(H(n)=h_1(n)\cdots h_g(n)\), where \(h_i\) are distinct irreducible polynomials each with integer coefficients and \(\deg h_i=k\) for all \(i=1,\ldots,g\). Suppose that \[ \sharp\{a \pmod{p} : (a,p)=1 \mbox{ and } H(a)\equiv 0 \pmod{p}\}<p-1. \] Then, for sufficiently large \(x\), there exists a natural number \(r\) such that \[ \sum\limits_{\substack{x<p\le 2x\\ \Omega(H(p))\le r}} 1 \gg \frac{x}{\log^{g+1} x}. \] If \(g\ge 2\) and \(k\) is sufficiently large, we may select an \(r\) of the form \[ r=gk+c_1g^{3/2}k^{1/2}+c_2g^2+O(g \log gk), \] where \(c_1\) and \(c_2\) are \(O(1)\). The authors also give explicit values of \(r\) for small \(g\) and \(k\). The case \(g=1\) was first investigated by \textit{H.E. Richert} [Mathematika 16, No. 1, 1--22 (1969; Zbl 0192.39703)] who showed that \(r=2k+1\) is an admissible choice. This was improved to \[ r=k+O(\log k) \] by \textit{A.J. Irving} [Bull. Lond. Math. Soc. 47, No. 4, 593--606 (2015; Zbl 1335.11077)]. For general \(g\), \textit{H. Halberstam} and \textit{H. E. Richert} [Sieve methods. London-New York San Francisco: Academic Press, a subsidiary of Harcourt Brace Jovanovich, Publishers. (1974; Zbl 0298.10026)] proved that \[ r=2gk+O(g\log gk) \] is admissible. The theorem above improves the last-mentioned result if the degree \(k\) is large compared to the number \(g\) of irreducible components. Its proof relies on an adaption of Irving's sieve method for the case \(g=1\) to the general case. Irving's novelty was to combine a one-dimensional with a two-dimensional sieve that permits a level of distribution beyond that available from the Bombieri-Vinogradov theorem. Similarly, in the case of general \(g\), the authors combine \(g\) and \((g+1)\)-dimensional sieves.