On the distribution of integers with a fixed number of prime factors
Let \(f_ k(n)\) be the characteristic function of the set of positive integers with exactly \(k\) different prime factors, \(\Pi(x,k)= \sum_{n\leq x} f_ k(n)\). The behaviour of \(\Pi(x,k)\) has been studied by numerous authors like Landau, Sathe, Selberg. In the critical range \(\log_ 2 x\ll k\ll \log x(\log_ 2 x)^{-2}\) (\(\log_ 2 x=\log\log x\)) very precise bounds were recently given by \textit{A. Hildebrand} and \textit{G. Tenenbaum} [Duke Math. J. 56, No. 3, 471-501 (1988; Zbl 0655.10036)]. In the present paper the authors prove a mean value theorem of Bombieri-Vinogradov type for \(f_ k(n)\), namely \[ \sum_{d\leq x^{1/2-\varepsilon}} \max_{(a,d)=1} \max_{y\leq x} \biggl| \sum_{\textstyle {{{n\leq y} \atop {n\equiv a(d)}}}} f_ k(n)- {1\over {\varphi(d)}} \sum_{\textstyle {{{n\leq y} \atop {(n,d)=1}}}} f_ k(n) \biggr|\;\ll\;\Pi(n,k)(\log x)^{-A} \] uniformly in the interval \(1\leq k\leq\eta\log x(\log_ 2 x)^{-2}\) (\(\eta=\eta(A,\varepsilon)>0\)). -- In the proof bounds for \(L\)-functions and zero density results are used.
distribution of integers with fixed number of prime factors
mean value theorem of Bombieri-Vinogradov
scientific article; zbMATH DE number 549784