The Median Largest Prime Factor

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.