The median largest prime factor

Let M(x)M(x) denote the median largest prime factor of the integers in the interval [1,x][1,x]. We prove thatM(x)=x1eexp(−lif(x)/x)+Oϵ(x1ee−c(logx)3/5−ϵ), where lif(x)=∫2x{x/t}logtdt. From this, we obtain the asymptoticM(x)=eγ−1ex1e(1+O(1logx)), where γ is the Euler–Mascheroni constant. This answers a question posed by Martin [3], and improves a result of Selfridge and Wunderlich [7].