Halász theorems for Gaussian ideals in sectors and short intervals
Abstract
We prove a quantitative Halász theorem for multiplicative functions on the nonzero ideals of ℤ[i], with bounds controlled by pretentious distance to the Archimedean characters Nit. 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 ℕ, and a theorem of Mangerel.
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BibTeX
@misc{kus2026halász,
title = {{Halász theorems for Gaussian ideals in sectors and short intervals}},
author = {Jan Kuś},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2605.19105},
eprint = {2605.19105},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.19105},
}
RIS
TY - UNPB
TI - Halász theorems for Gaussian ideals in sectors and short intervals
AU - Jan Kuś
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2605.19105
UR - https://arxiv.org/abs/2605.19105
AB - We prove a quantitative Halász theorem for multiplicative functions on the nonzero ideals of 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 ℕ, and a theorem of Mangerel.
ER -