<s>[INST] Solve and asymptotic expansion of $\sum_{a=1}^{H} \sum_{b=a+1}^{H} \left\lfloor{\frac{H}{a\, b}}\right\rfloor$ I am solving constrained polynomial systems resulting in constrained sums.  I am looking to see if $$\sum_{a=1}^{H} \sum_{b=a+1}^{H} \left\lfloor{\frac{H}{a\, b}}\right\rfloor$$ is expressible in terms of known functions and very importantly the asymptotic form as $H \rightarrow \infty$.  If needed we can make the sum over $b$ symmetric to the sum over $a$.
 [/INST] I get
$\frac14 n\ln^2(n)+O(n\ln(n))
$.
If
$\begin{array}\\
s(n)
&=\sum_{a=1}^{n} \sum_{b=a+1}^{n} \left\lfloor{\frac{n}{a\, b}}\right\rfloor\\
\text{then}\\
s(n)
&=\sum_{b=1}^{n} \sum_{a=1}^{b-1} \left\lfloor{\frac{n}{a\, b}}\right\rfloor\\
&=\sum_{a=1}^{n} \sum_{b=1}^{a-1} \left\lfloor{\frac{n}{a\, b}}\right\rfloor\\
\text{so}\\
2s(n)
&=\sum_{a=1}^{n} (\sum_{b=1}^{a-1} \left\lfloor{\frac{n}{a\, b}}\right\rfloor+ \sum_{b=a+1}^{n} \left\lfloor{\frac{n}{a\, b}}\right\rfloor)\\
&=\sum_{a=1}^{n} (\sum_{b=1}^{n} \left\lfloor{\frac{n}{a\, b}}\right\rfloor-\left\lfloor{\frac{n}{a^2}}\right\rfloor)\\
&=\sum_{a=1}^{n} \sum_{b=1}^{n} \left\lfloor{\frac{n}{a\, b}}\right\rfloor
-\sum_{a=1}^{n}\left\lfloor{\frac{n}{a^2}}\right\rfloor\\
&= u(n)-v(n)\\
v(n)
&= \sum_{a=1}^{n}\left\lfloor{\frac{n}{a^2}}\right\rfloor\\
&= \sum_{a=1}^{\lfloor \sqrt{n} \rfloor}\left\lfloor{\frac{n}{a^2}}\right\rfloor\\
&\le \dfrac{n\pi^2}{6}\\
\text{and}\\
u(n)
&=\sum_{a=1}^{n} \sum_{b=1}^{n} \left\lfloor{\frac{n}{a\, b}}\right\rfloor\\
&=\sum_{a=1}^{n} \sum_{b=1}^{\lfloor n/a \rfloor} \left\lfloor{\frac{n}{a\, b}}\right\rfloor\\
&=\sum_{a=1}^{n} \sum_{b=1}^{\lfloor n/a \rfloor} \left\lfloor{\frac{\lfloor n/a \rfloor}{b}}\right\rfloor\\
&\le \sum_{a=1}^{n} \lfloor n/a \rfloor\sum_{b=1}^{\lfloor n/a \rfloor} \left\lfloor{\frac1{b}}\right\rfloor\\
&\le \sum_{a=1}^{n} \lfloor n/a \rfloor\sum_{b=1}^{\lfloor n/a \rfloor} \frac1{b}\\
&\lt \sum_{a=1}^{n} \lfloor n/a \rfloor (\ln(n/a)+1)\\
&\le \sum_{a=1}^{n} (n/a)\ln(n/a)+ \sum_{a=1}^{n} (n/a)\\
&= \sum_{a=1}^{n} (n/a)\ln(n)-\sum_{a=1}^{n} (n/a)\ln(a)+ n\sum_{a=1}^{n} \frac1{a}\\
&= n\ln(n)\sum_{a=1}^{n} \frac1{a}-n\sum_{a=1}^{n} \frac{\ln(a)}{a}+ n\sum_{a=1}^{n} \frac1{a}\\
&= n\ln^2(n)+O(n\ln(n))-n(\frac12 \ln^2(n)+O(\ln(n))+ n(\ln(n)+O(1))\\
&= \frac12 n\ln^2(n)+O(n\ln(n))\\
\text{so}\\
s(n)
&\lt \frac14 n\ln^2(n)+O(n\ln(n))\\
\text{and}\\
u(n)
&=\sum_{a=1}^{n} \sum_{b=1}^{n} \left\lfloor{\frac{n}{a\, b}}\right\rfloor\\
&=\sum_{a=1}^{n} \sum_{b=1}^{\lfloor n/a \rfloor} \left\lfloor{\frac{n}{a\, b}}\right\rfloor\\
&\gt\sum_{a=1}^{n} \sum_{b=1}^{\lfloor n/a \rfloor} ({\frac{n}{a\, b}}-1)\\
&=\sum_{a=1}^{n} \sum_{b=1}^{\lfloor n/a \rfloor} {\frac{n}{a\, b}}-\sum_{a=1}^{n} \sum_{b=1}^{\lfloor n/a \rfloor}1\\
&=\sum_{a=1}^{n} \frac{n}{a}\sum_{b=1}^{\lfloor n/a \rfloor} {\frac1{b}}-\sum_{a=1}^{n} \lfloor n/a \rfloor\\
&>\sum_{a=1}^{n} \frac{n}{a}\ln(\lfloor n/a \rfloor)-n\ln(n)\\
&>\sum_{a=1}^{n} \frac{n}{a}(\ln( n/a)-1)-n\ln(n)\\
&= \frac12 n\ln^2(n)+O(n\ln(n))
\quad\text{by the same argument as above}\\
\end{array}
$
</s>