# Explicit estimates for the distribution of numbers free of large prime factors

## TL;DR

This paper provides explicit, tight bounds for counting smooth numbers, improving numerical understanding beyond traditional asymptotic estimates and challenging existing conjectures.

## Contribution

It introduces a saddle point method to derive explicit bounds for smooth numbers, surpassing previous asymptotic approximations.

## Key findings

- Explicit bounds for smooth number counts are tighter than previous estimates.
- The method can exclude the Dickman-de Bruijn asymptotic estimate as too small.
- The bounds challenge the Hildebrand-Tenenbaum main term as too large.

## Abstract

There is a large literature on the asymptotic distribution of numbers free of large prime factors, so-called $\textit{smooth}$ or $\textit{friable}$ numbers. But there is very little known about this distribution that is numerically explicit. In this paper we follow the general plan for the saddle point argument of Hildebrand and Tenenbaum, giving explicit and fairly tight intervals in which the true count lies. We give two numerical examples of our method, and with the larger one, our interval is so tight we can exclude the famous Dickman-de Bruijn asymptotic estimate as too small and the Hildebrand-Tenenbaum main term as too large.