Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2603.17682 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912972696190976 |
|---|---|
| author | Verwee, Johann |
| author_facet | Verwee, Johann |
| contents | In previous joint work with Tenenbaum, the truncation step $f \mapsto f_R$ in the conditional effective Erdos-Wintner theorem on the fibre $ω(n)=k$ yields, in the continuous case for real strongly additive $f$, a remainder of size $η_f(R)^{r/(r+1)}$, 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 $κ\le r \le 1/κ$, this bound improves to the natural linear scale $rη_f(R)$ 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, $Ω$-fibres, and weighted fibres whenever the corresponding ratio estimate is available. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_17682 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Linear truncation for conditioned prime-factor fibres Verwee, Johann Number Theory 11N25, 11N60 In previous joint work with Tenenbaum, the truncation step $f \mapsto f_R$ in the conditional effective Erdos-Wintner theorem on the fibre $ω(n)=k$ yields, in the continuous case for real strongly additive $f$, a remainder of size $η_f(R)^{r/(r+1)}$, 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 $κ\le r \le 1/κ$, this bound improves to the natural linear scale $rη_f(R)$ 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, $Ω$-fibres, and weighted fibres whenever the corresponding ratio estimate is available. |
| title | Linear truncation for conditioned prime-factor fibres |
| topic | Number Theory 11N25, 11N60 |
| url | https://arxiv.org/abs/2603.17682 |