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 $\ddot{I}‰(n)=k$ yields, in the continuous case for real strongly additive $f$, a remainder of size $\hat{I}{\cdot }_f(R{)}^{r/(r+1)}$, 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 $\hat{I}º\le r\le 1/\hat{I}º$, this bound improves to the natural linear scale $r\hat{I}{\cdot }_f(R)$ 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, $\hat{I}\text{c◯}$-fibres, and weighted fibres whenever the corresponding ratio estimate is available.

10 pages, no figures