Linear truncation for conditioned prime-factor fibres

Abstract

In previous joint work with Tenenbaum, the truncation step $f \mapsto f_R$ in the conditional effective Erdos-Wintner theorem on the fibre $\omega(n)=k$ yields, in the continuous case for real strongly additive $f$, a remainder of size $\eta_f(R)^{r/(r+1)}$, where $R$ is the truncation level and $r=k/\log\log x$. We prove an effective linear truncation lemma showing that, in the central window $\kappa \le r \le 1/\kappa$, this bound improves to the natural linear scale $r\eta_f(R)$ under an effective Sathe-Selberg-type ratio estimate for the fibre. This yields a direct effective sharpening of the truncation step in the previous joint work. The same truncation upgrade also applies to prime-set restrictions, $\Omega$-fibres, and weighted fibres whenever the corresponding ratio estimate is available.