I am interested in determining an asymptotic formula for the double summation of $1/(ab)$, where $a$ is an odd integer ranging between 1 and $k/\sqrt{j}$, $b$ is an odd integer ranging between $a$ and $aj$, $j$ is a real number $>1$, and $k$ tends to infinity. In symbols,
$$ \sum_{\substack{1 \leq a \leq k/\sqrt{j} \\ a \text{ odd}}} \sum_{\substack{a \leq b \leq aj \\ b \text{ odd}}} \frac{1}{ab}. $$
For $j=1$, the result of the summation simply corresponds to the infinite harmonic sum of odd squares $1/1 + 1/9 + 1/25\ldots $, which yields $\pi^2/8$.
For $j>1$, I obtained the formula $\tfrac{1}{4} \ln(k) \ln(j) + O(1)$. I am particularly interested in this $O(1)$ term. Plotting this term vs $j$ we obtain a discontinuous function, where the most evident discontinuities occur when $j$ is an odd integer. For instance, setting $j=2$, the constant term is about $0.94$. It progressively decreases (with other discontinuities) to approximately $0.73$ as $j$ increases approaching $3$, but for $j=3$ it raises to about $1.14$. The abrupt increase observed for $j=3$ is equal to $\pi^2/24$ (and more generally, for any odd integer $j$, the term shows a discontinuity with an abrupt increase by $\pi^2/8/j$).
Is there any way to express the values of this $O(1)$ term explicitly? Thank you.