On the Barban-Davenport-Halberstam theorem. XIV (Q2773335)

This paper is concerned with Barban-Davenport-Halberstam type theorems for general sequences. Consider an increasing sequence \(s\) of positive integers, and let NEWLINE\[NEWLINES(x;a,k) := \sum_{\substack{ s\leq x\\ s \equiv a\pmod k}} 1 .NEWLINE\]NEWLINE Suppose this sequence is well distributed in arithmetic progressions so that we can write \(S(x;a,k) = f(a,k) x + E(x;a,k)\), where \(E(x;a,k)\) is an error term. Let NEWLINE\[NEWLINEH(x,k) = \sum_{0<a\leq k}E^2(x;a,k), \qquad G(x,Q) = \sum_{k\leq Q}H(x,k).NEWLINE\]NEWLINE Under various assumptions, the author has in previous papers obtained the asymptotic formula NEWLINE\[NEWLINE G(x,Q) = (D_1 +o(1))Qx +O(x^2 \log ^{-A} x),NEWLINE\]NEWLINE which is of interest in the range \(x\log^{-A}x \leq Q\leq x\). The situation when \(D_1=0\) is studied in this paper and occurs when \(E(x;a,k)\) is sufficiently small. The author supposes that \(s\) satisfies \( |E(x;a,k) |< A_2({x\over k})^\alpha \) for \(k\leq x^{1/2}\), \(A_2\) is a positive constant, and \(0<\alpha < 1/2\) a constant; this is referred to as Criterion \(V_1\). With this condition the author proves, using a form of the circle method, that \( G(x,Q) \ll Q^{2-2\alpha}x^{2\alpha}\log^2(2x/Q)\) for \(Q>x^{3/4}\log^2 x\). Stronger criteria allow one to eliminate the log factor above. Thus one sees that a strong enough estimate for \(E(x;a,k)\) for \(k\leq x^{1/2}\) implies that almost the same estimate will frequently continue to hold for much larger values of \(k\). NEWLINENEWLINENEWLINEThe method also obtains the interesting asymptotic formula, for \(0<c<x\), NEWLINE\[NEWLINE \sum_{\substack{ x<s,s'\leq 2x\\ s-s'=c}} 1 = (x-c){\mathfrak S}(c) +O(x^{1/2+\alpha}\log^2x)NEWLINE\]NEWLINE as \(x\to \infty\), where \({\mathfrak S}(c)\) is the singular series associated with \(s\). Thus binary additive problems can be handled by the circle method for sequences which satisfy Criteria \(V_1\). NEWLINENEWLINENEWLINEThe author obtains the following result on \(H(x,k)\) assuming both Criterion \(V_1\) and that \(f(a,k)=g(k,(a,k))\). Let \((3/2-\alpha)/(2-2\alpha) < \beta <1\) and suppose \(k\) is a prime number between \(x^\beta\) and \(x\). Then \(H(x,k)= O(k^{1-2\alpha}x^{2\alpha}(x/k)^\epsilon)\). With a stronger criterion the \(\epsilon\) factor can be replaced by \(\log^5(2x/k)\). NEWLINENEWLINENEWLINEAs an example, the author constructs a sequence which is well distributed in arithmetic progressions but has an oscillatory main term. NEWLINENEWLINENEWLINERecently \textit{R. C. Vaughan} [Philos. Trans. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 356, 781-791, 793-809 (1998; Zbl 0902.11039, Zbl 0902.11040)] has obtained results related to the results of this paper.