A conditional resolution of the parity problem in sieve theory (Q1183275)

Let \(T(x,\alpha)\) be the number of primes \(p\) with \(p\leq x\), \(p + 2=p_1p_2p_3\), and \(p_i\geq x^\alpha\) for \(1\leq i \leq 3\). Let \(C=\prod_{p>2} (1-(p-1)^{-2})\) be the twin prime constant and \[ I=\int_{\alpha}^{1-2\alpha} \log((1-t)/\alpha -1)t^{-1}(1-t)^{- 1}\, dt. \] The conditions of the title are (1) the Elliott-Halberstam conjecture and (2) an estimate of the form \(T(x,\alpha)\leq \beta ICx \log^{-2}x\). (The Elliott-Halberstam conjecture alone gives (2) with \(\beta=2/3 + \varepsilon\).) The author shows that if (1) and (2) are true then \[ |\{p\leq x: p + 2=p_1p_2\}|\geq (2-3\beta- \varepsilon)ICx\log^{-2}x. \]