The asymptotic distribution of the largest prime divisor

TL;DR

This paper investigates the asymptotic behavior of the largest prime divisor of integers, deriving a formula involving iterated logarithms in a specific limiting regime.

Contribution

It provides a new asymptotic expression for the Dickman function in a particular limit, connecting it to iterated logarithms.

Findings

Derived asymptotic formula for the Dickman function in the specified limit

Expressed the asymptotic value using ratios of iterated logarithms

Enhanced understanding of the distribution of large prime divisors

Abstract

We consider the Dickman function $\Psi (x,y)$ in the limit when $\ln x/\ln y\to \infty $ and $\ln \ln x \ln y\to 0$. The asymptotic value is expressed in terms of the ratio of iterated loragithm of $x$ and $ln y$.