Explicit bounds for Dickman's function

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.

10 pages, 1 figure