TITLE: Median largest-prime-factor
QUESTION [16 upvotes]: Let $P(n)$ denote the largest prime factor of $n$. For any integer $x\ge2$, define the median
$$
M(x) = \text{the median of the set }\{P(2), P(3), \dots, P(x) \}.
$$
Classical results of Dickman and de Bruijn show that the median is roughly $x^{1/\sqrt{e}}$. More specifically, I think that the Dickman-de Bruijn rho-function approach can show the following: for any function $f(x)$ tending to infinity with $x$, the median $M(x)$ is between $x^{1/\sqrt{e}}/f(x)$ and $x^{1/\sqrt{e}}f(x)$ for all sufficiently large $x$.
But I got to thinking the other day: is there a way to determine how the median compares to $x^{1/\sqrt{e}}$ specifically? In other words, which one of the following is true?

 For all sufficiently large $x$, we have $M(x) \lt x^{1/\sqrt{e}}$.
 Each inequality $M(x) \lt x^{1/\sqrt{e}}$ and $M(x) \ge x^{1/\sqrt{e}}$ holds for arbitrarily large $x$.
 For all sufficiently large $x$, we have $M(x) \ge x^{1/\sqrt{e}}$.

REPLY [9 votes]: I posted a paper to the arXiv which deals with this question along with some other things.
Using some results regarding either the mean of $\omega(n)$, or integers without large prime factors, we can prove that $$M(x)=e^{(\gamma-1)/\sqrt{e}}x^{1/\sqrt{e}}\left(1+O\left(\frac{1}{\log x}\right)\right),$$ where $\gamma$ is the Euler Mascheroni constant.
More specifically, if we let $\text{li}_f(x)=\int_2^x \frac{\{x/t\}}{\log t} dt$, then $$M(x)=x^{\frac{1}{\sqrt{e}}\exp\left(-\frac{\text{li}_f(x)}{x}\right)}\left(1+O\left(e^{-c\sqrt{\log x}}\right)\right).$$  (In case the TeX is not readable, the exponent of $x$ in the above equation is $\frac{1}{\sqrt{e}}\exp\left(-\frac{\text{li}_f(x)}{x}\right)$.)  
This function $\text{li}_f(x)$ has the asymptotic expansion $$\text{li}_f(x)=c_0 \frac{x}{\log x}+c_1 \frac{x}{\log^2 x}+\cdots+c_{k-1} \frac{(k-1)!x}{\log^k x}+O\left(\frac{x}{\log^{k+1}(x)}\right),$$
where $$c_k=1-\sum_{j=0}^k\frac{\gamma_k}{k!},$$ and $\gamma_k$ denotes the $k^{th}$ Stieltjes constant.