Halász theorems for Gaussian ideals in sectors and short intervals
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911696638967808 |
|---|---|
| author | Kuś, Jan |
| author_facet | Kuś, Jan |
| contents | We prove a quantitative Halász theorem for multiplicative functions on the nonzero ideals of $\mathbb{Z}[i]$, with bounds controlled by pretentious distance to the Archimedean characters $N^{it}$. We also prove a sectorial analogue: under angular non-pretentiousness, the sum of $f$ over ideals lying in a fixed sector is asymptotically given by the expected proportion of the unrestricted sum. Finally, under angular non-pretentiousness and a non-degeneracy condition on conjugate prime pairs, we prove a sectorial short-interval version of the Halász theorem for annular sectors whose radial thickness tends to infinity. The proof of the sectorial short-interval Halász theorem uses angular Fourier expansion, norm-compression to multiplicative functions on $\mathbb{N}$, and a theorem of Mangerel. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_19105 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Halász theorems for Gaussian ideals in sectors and short intervals Kuś, Jan Number Theory 11N37 (Primary), 11N60, 11R11 (Secondary) We prove a quantitative Halász theorem for multiplicative functions on the nonzero ideals of $\mathbb{Z}[i]$, with bounds controlled by pretentious distance to the Archimedean characters $N^{it}$. We also prove a sectorial analogue: under angular non-pretentiousness, the sum of $f$ over ideals lying in a fixed sector is asymptotically given by the expected proportion of the unrestricted sum. Finally, under angular non-pretentiousness and a non-degeneracy condition on conjugate prime pairs, we prove a sectorial short-interval version of the Halász theorem for annular sectors whose radial thickness tends to infinity. The proof of the sectorial short-interval Halász theorem uses angular Fourier expansion, norm-compression to multiplicative functions on $\mathbb{N}$, and a theorem of Mangerel. |
| title | Halász theorems for Gaussian ideals in sectors and short intervals |
| topic | Number Theory 11N37 (Primary), 11N60, 11R11 (Secondary) |
| url | https://arxiv.org/abs/2605.19105 |