Abstract
We combine A.J. Irving's sieve method with the Diamond-Halberstam-Richert (DHR) sieve to study the almost-prime values produced by products of irreducible polynomials evaluated at prime arguments. A significant improvement is made when the degrees of the irreducible components are large compared with the number of components themselves. This generalizes the previous results of Irving and Kao, who separately examined the almost-prime values of a single irreducible polynomial evaluated at prime arguments.
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