On the joint distribution of the two ``number of prime factors'' functions. (Q1106881)

Let \(V(x,\lambda)=\#\{n\leq x: \Omega (n)>\lambda \omega (n)\}\), where \(\Omega(n)\) and \(\omega(n)\) denote the number of prime factors of \(n\) counted with, resp. without, multiplicity. An asymptotic formula for \(V(x,\lambda)\), valid for every \(\lambda >1\), has been given recently by \textit{M. Balazard} [Sur la moyenne des exposants dans la décomposition en facteurs premiers, Acta Arith. 52, No. 1, 11--23 (1989; Zbl 0682.10031)]. In the paper under review, the author obtains sharper estimates by a different method: If \(\lambda =a/q\), \((a,q)=1\), is rational, then \[ V(x,\lambda)=\frac{\log 2}{q(2^{1/q}-1)}H(\alpha)\;x(\log x)^{\alpha -1}\;\Bigl(1+O_{\varepsilon}\Bigl((\log x)^{-\gamma +\varepsilon}\Bigr)+O\Bigl(\log q (\log x)^{-2\gamma \sin 2(\pi /q)}\Bigr)\Bigr) \] holds with \(\alpha =2^{1-\lambda}\), \(\gamma =2^{1-\lambda}-3^{1- \lambda}\) and \[ H(\alpha)=\frac{\alpha 2^{-\alpha}}{\Gamma (\alpha)\log 4}\prod_{p>2}\Bigl(1-\frac 1p\Bigr)^{\alpha}\;\Bigl(1+\frac{\alpha}{p-2}\Bigr) \] uniformly for \(x\geq e^q\) and \(1<\lambda_ 1\leq \lambda \leq \lambda_ 2\). If \(\lambda\) is irrational, a similar estimate holds with main term \(H(\alpha) x(\log x)^{\alpha -1}\) and an error term that depends on Diophantine approximation properties of \(\lambda\).