TITLE: Bounds re Asymptotic Formula for the Sum of Largest Prime Factors
QUESTION [6 upvotes]: I have a reference request related to the result :
$\sum_{n=2}^{x} P(n)$ ~ $\frac{\pi^2}{12}\frac{x^{2}}{log(x)}$ as $x \rightarrow \infty$
where $P(n)$ is the largest prime factor of the positive integer $n$.
Theorem 1.1 in [1] below appears to be the first proof of this result and the result was generalized in Theorem 3.1 in [2] below. 
I am looking for upper and lower bounds on the ratio L/R where L and R are the left and right sides respectively of the asymptotic relation above.
Thanks for any help.
References
[1]. K.Alladi and P.Erdos. Pacific J. Math. 71(1977) 275-294
[2]. J.De Konnick and R.Sitaramachandrarao. Indian J. Pure Appl Math. 19(10) 990-1004 Oct 1988
Disclosure: This question was first posted on Math Stack Exchange where it has languished for the last three weeks without an answer (or even a comment).

REPLY [6 votes]: I am not exactly sure what you mean by bounds for $L/R$, but I assume that this translates to the rate of convergence and the next terms in the asymptotic. In this short note, a more precise asymptotic for $\sum_{n\leq x} P(n)$ is given and it is shown that $$\frac{1}{x}\sum_{n\leq x}P(n)=\text{li}_g(x) +O_\epsilon \left(x e^{-c(\log x)^{3/5-\epsilon}}\right),$$ where $$\text{li}_g(x)=\int_2^x \frac{t}{x}\frac{[x/t]}{\log t}dt$$ is an integral function that shares some properties in common with $\text{li}(x)=\int_2^x \frac{1}{\log t}dt.$ In particular, we have the asymptotic expansion 
$$\text{li}_g(x)= \frac{c_0 x}{\log x}+\frac{1!c_1 x}{\log^2x}+\cdots+\frac{(k-1)!c_{k-1} x}{\log^{k}x}+O_k\left(\frac{x}{\log^{k+1}x}\right)$$
where $$c_k=\frac{1}{2^{k+1}}\sum_{j=0}^n \frac{2^j(-1)^j\zeta^{(j)}(2)}{j!},$$ which yields an asymptotic expansion for $\sum_{n\leq x} P(n).$