Montgomery's weighted sieve for dimension two (Q1232433)

Montgomery's weighted sieve (see [\textit{H. L. Montgomery}, Topics in multiplicative number theory, Lect. Notes Math. 227. Berlin: Springer (1971; Zbl 0216.03501); \textit{H. L. Montgomery} and \textit{R. C. Vaughan}, Mathematika 20, 119--134 (1973; Zbl 0296.10023)]) is used to prove the following theorem: let \(a\neq 0, b\neq 0\)be integers with \((a,b)=1\), \(2\mid ab\). Then we have for \(x>1\) \[ \sum_{_{\substack{ p\leq x\\ ap+b\in\mathcal P}}} 1\leq 16\omega \frac x{\log^2x}\prod_{_{\substack{ p\mid ab\\ p>2}}} \frac{p-1}{p-2},\quad \omega:=\prod_{p>2} (1-(p-1)^{-2}), \] (\(\mathcal P\) denotes the set of all prime numbers). Recently, \textit{R. C. Vaughan} [J. Reine Angew. Math. 290, 93--108 (1977; Zbl 0344.10028)] used an estimate like this to prove that every even number \(n\geq 2\) is the sum of at most 26 primes.