TITLE: Average $lcm(a,b)$, $ 1\le a \le b \le n$, and asymptotic behavior
QUESTION [5 upvotes]: What is the average value for $\mathrm{lcm}(a,b)$, with $ 1\le a \le b \le n$, for a
  given $n$, and what is the asymptotic behavior? The $\mathrm{lcm}$ is the least common multiple.

I have calculated, as $(n, avg)$: 
$$(10, 19.836)$$
$$(100, 1826.859)$$
$$(1000, 182828.976)$$
The values appear to converge quadratically to some constant.
This generalizes to $$\sum_{1 \le j \le k \le n} \mathrm{lcm}(j,k)$$

REPLY [3 votes]: Theorem 6.3 of Olivier Bordelles, Mean values of generalized gcd-sum and lcm-sum functions, Journal of Integer Sequences, Vol. 10 (2007), Article 07.9.2, says, for any real number $x$ sufficiently large, $$\sum_{n\le x}\sum_{j=1}^n[n,j]={\zeta(3)\over8\zeta(2)}x^4+O\bigl(x^3(\log x)^{2/3}(\log\log x)^{4/3}\bigr)$$