My guess is that there exists a constant $C$ such that $A(X) \sim C (\log X)^2$.



Is there a known asymptotic for $A(X):= \sum_{1 \leq i,j \leq X} \frac{1}{\mathrm{lcm}(i,j)}$?
$$ \sum\_{1 \leq i,j \leq X} \frac{1}{\mathrm{lcm}(i,j)} = \sum\_{1 \leq i,j \leq X} \frac{\mathrm{gcd}(i,j)}{ij} $$
$$ = \sum\_{1 \leq i,j \leq X} \frac{\sum\_{d|i,j} \phi(d)}{ij}$$
$$ = \sum\_{d \leq X} \phi(d) \sum\_{1 \leq i,j \leq X: d|i,j} \frac{1}{ij}$$
$$ = \sum\_{d \leq X} \frac{\phi(d)}{d^2} \sum\_{1 \leq i',j' \leq X/d} \frac{1}{i'j'}$$
$$ = \sum\_{d \leq X} \frac{\phi(d)}{d^2} ( \sum\_{1 \leq i \leq X/d} \frac{1}{i})^2$$
$$ = \sum\_{d \leq X} \frac{\phi(d)}{d^2} ( \log(X/d) + O(1))^2$$
$$ = \sum\_{d \leq X} \frac{\phi(d)}{d^2} ( \log^2(X) - 2 \log(d) \log(X) + \log^2(d) + O( \log X ) )$$
$$ = A\_0 \log^2 X - 2 A\_1 \log X + A\_2 + O( \log^2 X )$$


where
$$ A\_j := \sum\_{d \leq X} \frac{\phi(d) \log^j d}{d^2}.$$


One can compute asymptotics for the $A\_j$ by Perron's formula, but we proceed instead by elementary means. Since $\phi(d) = \sum\_{d=ab} \mu(a) b$ we have
$$ A\_j = \sum\_{ab \leq X} \frac{\mu(a) b \log^j(ab)}{a^2b^2}$$
$$ = \sum\_{a \leq X} \frac{\mu(a)}{a^2} (\frac{\log^{j+1}(X)}{j+1} + O( (1 + \log^j(a)) \log^j X ) )$$
$$ = \frac{1}{j+1} \log^{j+1} X \sum\_{a \leq X} \frac{\mu(a)}{a^2} + O(\log^j X)$$
$$ = \frac{1}{(j+1)\zeta(2)} \log^{j+1} X + O(\log^j X).$$
Hence
$$ \sum\_{1 \leq i,j \leq X} \frac{1}{\mathrm{lcm}(i,j)} = \frac{1}{3 \zeta(2)} \log^3 X + O(\log^2 X).$$


