On the distribution in the arithmetic progressions of reducible quadratic polynomials in short intervals. (Q5933532)

Let \(\mathcal A\) denote the sequence of numbers \(n(n+2)\) with \(x<n<x+x^ \theta\), and write \(| {\mathcal A}_ d| \) for the number of these that are divisible by an odd squarefree number~\(d\). Then it is entirely elementary to obtain an asymptotic estimate for \(\sum_ {d<D}\left| {\mathcal A}_ d \right| \) when \(D = x^ {\theta-\varepsilon}\). Following the work of \textit{H.~Iwaniec} [Acta Arith. 37, 307--320 (1980; Zbl 0444.10038)] on the bilinear form of the remainder term in the linear sieve one might be interested in \(\sum_ {M<m \leq 2M}\sum_ {N < n \leq 2N} \alpha_ m\beta_ n\left| {\mathcal A}_ {mn}\right| \), where \(a_ m\) and~\(b_ n\) are bounded, for example by~\(1\). In the current instance the associated sifting situation would, however, not be ``linear''. Nevertheless, subject to the requirement that an asymptotic estimate for this expression can be obtained, one might like to maximise the product~\(MN\). At least one would like to improve on the ``elementary'' value obtained by using \(M=D\), \(N=1\). NEWLINENEWLINEThe authors consider this problem subject to the extra condition that the integers \(m\) and~\(n\) are restricted to be prime. Then they obtain a result of the required type with a certain \(M\) and~\(N\) satisfying \(MN=x^ {3\theta-3/2- \varepsilon}\), which remains significant (because \(MN \leq D\)) when \({3\over4}<\theta<1\). They state a generalisation in which \(n(n+2)\) is replaced by a product of two linear polynomials. NEWLINENEWLINETheir proof uses the large sieve, and improves on the result at \(\theta=1\) derived by \textit{S. Salerno} and \textit{A.~Vitolo} [Russ. Acad. Sci., Izv., Math. 45, No.~1, 215--228 (1995) and Izv. Ross. Akad. Nauk, Ser. Mat. 58, No.~4, 211--223 (1994; Zbl 0839.11040)] using Weil's estimates for Kloosterman sums.