On a new approach to selecting the parameters of a two-dimensional sieve with weights (Q1898278)

The author considers a typical two-dimensional sieve problem. Let \(F\) be the product of two irreducible polynomials, \(\deg F= q\geq 3\). It is shown that \(F(n)\) is a \(P_r\) (a number with \(\leq r\) prime factors) for infinitely many \(n\) if \(r> q+\ln \ln q+ 7,27\). This is a slight improvement of a result of \textit{H. Halberstam} and \textit{H.-E. Richert} [Sieve methods (1974; Zbl 0298.10026)]. The proof uses Richert's weights. The improvement is achieved by a careful study of an integral occurring in the work of Halberstam and Richert. A similar improvement is gained for the problem of finding infinitely many primes \(p\) for which \(F(p)\) is a \(P_r\).