The Median Largest Prime Factor

TL;DR

This paper derives an asymptotic formula for the median of the largest prime factors of integers up to x, providing a precise growth rate and answering a previously posed question.

Contribution

It introduces a new asymptotic expression for the median largest prime factor, improving upon earlier results and resolving an open question.

Findings

Asymptotic formula for M(x) involving exponential and logarithmic terms

Confirmation of the median largest prime factor's growth rate as a power of x

Improvement over previous bounds and results by Selfridge and Wunderlich

Abstract

Let $M(x)$ denote the median largest prime factor of the integers in the interval $[1,x]$. We prove that $$M(x)=x^{\frac{1}{\sqrt{e}}\exp(-\text{li}_{f}(x)/x)}+O_{\epsilon}(x^{\frac{1}{\sqrt{e}}}e^{-c(\log x)^{3/5-\epsilon}})$$ where $\text{li}_{f}(x)=\int_{2}^{x}\frac{\{x/t\}}{\log t}dt$. From this, we obtain the asymptotic $$M(x)=e^{\frac{\gamma-1}{\sqrt{e}}}x^{\frac{1}{\sqrt{e}}}(1+O(\frac{1}{\log x})),$$ where $\gamma$ is the Euler Mascheroni constant. This answers a question posed by Martin, and improves a result of Selfridge and Wunderlich.