On the A-dependence of the linear sieve with application to polynomial sequences
This paper is concerned with error terms in the linear sieve. In the notation of \textit{H. Halberstam} and \textit{H.-E. Richert}'s book ''Sieve Methods'' (1974; Zbl 0298.10026), the linear sieve gives the bounds \[ S(A,P,z)\leq XV(z) \{F(u)+H\}+\sum_{d<y,\;d | P(z)}| R_ d|, \] \[ S(A,P,z)\geq XV(z) \{f(u)-H\}+\sum_{d<y,\;d | P(z)}| R_ d|. \] Here, \(H\) is an error term which depends on the constant \(A_ 2\) in the hypothesis \[ V(v)/V(w)\leq (\log v/\log w)(1+A_ 2/\log v),\;2\leq v\leq w. \] (This is hypothesis \(\Omega_ 2(1,A_ 2)\) of ''Sieve methods''.) Halberstam and Richert proved the estimate \(H\ll_{A_ 2} (\log X)^{-1/14}.\) \textit{H. Iwaniec} [Acta Arith. 36, 171--202 (1980; Zbl 0435.10029)] was the first to give an estimate for \(H\) with the dependence on \(A_ 2\) made explicit; he proved that \[ H=(\log y)^{-1/3}\;\exp (A_ 2^{1/2}-u \log u+u \log \log 3u+O(u)). \] Richert (unpublished) improved this to \(H\ll A_ 2^ 8 e^{-u}(\log z)^{-1/3}\). In this paper, the author replaces \(\Omega_ 2(1,A_ 2)\) with \((\Omega_ a)\): There exists a constant \(A\geq 2\) such that \[ V(v)/V(w)\leq (\log v/\log w)(1+A/\log v)\quad\text{if}\;a\leq v<w. \] With this assumption, he proves that \[ H \ll_ a V^{-1} A^{4+\alpha} (\log A)^{2+\alpha} (\log z)^{-\alpha}\quad \text{for}\;0<\alpha \leq 1/3. \] As an application, the author gives upper bounds on the number of primes in polynomial sequences. Unlike previous bounds of this sort, his bounds are uniform in the discriminant of the corresponding polynomial.
number of primes in polynomial sequences
scientific article; zbMATH DE number 3972981
scientific article