The Median Largest Prime Factor

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.

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