One-sided sifting density hypotheses in Selberg's sieve (Q2710117)

Following the formulation and notation of the book ``Sieve Methods'' by \textit{H. Halberstam} and \textit{H.-E. Richert} (Academic Press) (1974; Zbl 0298.10026), we let \(\mathcal A\) be a finite sequence of positive integers, \(X\) an associated real number, and \(\omega\) a multiplicative function with \(0\leq \omega(p) <p\). The quantity \(X\) and the function \(\omega\) are chosen so that the quantity \({\mathcal A}_d\) is well approximated by \(X \omega(p)/p\). Applying Selberg's upper bound sieve requires estimates for the quantity \(G(z)=\sum_{d\leq z} \mu^2(d) g(d)\), where \(g(d)=\prod_{p|d} \omega(p)/(p-\omega(p))\). Halberstam and Richert (op cit.) gave an asymptotic estimate for \(G(z)\) subject to the hypothesis that NEWLINE\[NEWLINE-L \leq \sum_{u\leq p < v} g(p) \log p - \kappa \log (v/u) \leq ANEWLINE\]NEWLINE when \(2\leq u < v\). For some applications, it is desirable to have a lower bound for \(G(z)\) subject to the one-sided hypothesis NEWLINE\[NEWLINE\sum_{u\leq p < v} g(p)\log p \leq \kappa \log(v/u) + A. \tag{1}NEWLINE\]NEWLINE \textit{D. A. Rawsthorne} [Acta Arith. 41, 281-299 (1982; Zbl 0486.10031)] proved a lower bound for \(G(z)\) subject to (1). In this paper, the authors give an alternate proof that is more direct and stronger than that of Rawsthorne. They prove that if (1) is true, then NEWLINE\[NEWLINEG(z)\geq e^{-\gamma} \kappa (\Gamma(\kappa+1) W(z))^{-1} (1+O(A/\log z)),NEWLINE\]NEWLINE where the implied constant depends only on \(\kappa\) and \(W(z)=\prod_{p<z} ( 1 - \omega(p)/p).\) Rawsthorne's result had an extra power of \(\log \log z\) in the error term.NEWLINENEWLINEFor the entire collection see [Zbl 0959.00053].