PreprintCC BY 4.0arXiv
Omega bounds for the Piltz divisor problem
Abstract
We improve our earlier omega bounds for the error term in the Piltz divisor problem over number fields. Our proof uses a weighted extension of a counting theorem of Balasubramanian and Ramachandra and an extension of Lamzouri's kernel to apply the long resonance method. We also discuss applications of our results to lattice points in norm-form regions, invariant sublattices, abelian covers of curves, spectral counting, and representation growth of Heisenberg groups.
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BibTeX
@misc{karak2026omega,
title = {{Omega bounds for the Piltz divisor problem}},
author = {Nilmoni Karak and Kamalakshya Mahatab},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.33820},
eprint = {2609.33820},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33820},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Omega bounds for the Piltz divisor problem
AU - Nilmoni Karak
AU - Kamalakshya Mahatab
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2609.33820
UR - https://arxiv.org/abs/2609.33820
AB - We improve our earlier omega bounds for the error term in the Piltz divisor problem over number fields. Our proof uses a weighted extension of a counting theorem of Balasubramanian and Ramachandra and an extension of Lamzouri's kernel to apply the long resonance method. We also discuss applications of our results to lattice points in norm-form regions, invariant sublattices, abelian covers of curves, spectral counting, and representation growth of Heisenberg groups.
ER -