On a problem of A. Rényi (Q1181779)

Let as usual \(\omega(n)\) and \(\Omega(n)\) denote the number of distinct prime factors of \(n\) and the number of all prime factors of \(n\), respectively. \textit{A. Rényi} [Publ. Inst. Math. Acad. Serbe Sci. 8, 157--162 (1955; Zbl 0066.03203)] was the first to prove that, for fixed \(q\geq 0\) an integer and suitable \(d_q\geq 0\) one has, as \(x\to\infty\), \[ F_q(x)=\sum_{n\leq x,\Omega(n)-\omega(n)=q}1=d_qx+o(x). \] Later this result was improved several times [see e.g. Ch. 14 of the reviewer's book ``The Riemann zeta-function.'' New York: John Wiley (1985; Zbl 0556.10026)], most notably by H. Delange. The author makes a significant contribution to the study of \(F_q(x)\) by proving the following result, which supersedes all the previous ones in sharpness: Let \(N(x)=(\log x)^{1/3}(\log\log x)^{-1/3}\), and let \(b_j\) denote positive absolute constants. Then uniformly for integers \(q\) such that \(0\leq q\leq b_1 N(x)\) one has \[ F_q(x)=d_qx+x^{1/2}\sum_{1\leq j\leq N(x)}P_{j,q}(\log\log x)(\log x)^{-j-1}+O(x^{1/2}\exp(-b_2N(x))), \] where the polynomials \(P_{j,q}\) (of degree \(\leq q-1)\) satisfy \[ | P_{j,q}(\log\log x)|\leq b_3b^j_4(j+1)! q\log x\quad(x\geq 10). \] The proof is by complex integration, making use of a loop contour (at 1/2) and the Vinogradov-Korobov zero-free region for \(\zeta(s)\) (see Ch. 6, op. cit.).