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arXiv · 2606.07785

Explicit bounds for Dickman's function

Andreas Weingartner
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Abstract

Dickman's function $\rho(u)$ 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 $\rho(u)$, resulting in estimates with a relative error of less than $0.005/u^2$ for all $u\ge 5$. This allows for an approximate evaluation of $\rho(u)$ without the need to solve the delay differential equation numerically.

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Andreas Weingartner. 2026-06-05. Explicit bounds for Dickman's function. https://arxiv.org/abs/2606.07785

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