PreprintCC BY 4.0arXiv
Explicit bounds for Dickman's function
Abstract
Dickman's function ρ(u) denotes the natural density of integers n whose largest prime factor does not exceed n1/u. We establish numerically explicit upper and lower bounds for ρ(u), resulting in estimates with a relative error of less than 0.005/u2 for all u≥ 5. This allows for an approximate evaluation of ρ(u) without the need to solve the delay differential equation numerically.
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BibTeX
@misc{weingartner2026explicit,
title = {{Explicit bounds for Dickman's function}},
author = {Andreas Weingartner},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.07785},
eprint = {2606.07785},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.07785},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Explicit bounds for Dickman's function
AU - Andreas Weingartner
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2606.07785
UR - https://arxiv.org/abs/2606.07785
AB - Dickman's function ρ(u) denotes the natural density of integers n whose largest prime factor does not exceed n^1/u. We establish numerically explicit upper and lower bounds for ρ(u), resulting in estimates with a relative error of less than 0.005/u^2 for all u≥ 5. This allows for an approximate evaluation of ρ(u) without the need to solve the delay differential equation numerically.
ER -