The probability distribution of LCM of uniformly distributed integers in {1,…,n}\{1,\ldots,n\}

In the recent paper by Fernandez and Fernandez here on ArXiv, the following formula which was first proved by Diaconis and Erdos appears, on page 2.

For $0<t\leq 1$ the distribution of the lcm of independent pairs of integers $X_1,X_2$ uniformly drawn from $\{1,\ldots,n\}$ satisfies:
$$\mathbb{P}(\mathrm{lcm}(X_1,X_2)\leq t n^2)=1-\frac{1}{\zeta(2)} \sum_{j=1}^{\lfloor \frac{1}{t}\rfloor} \frac{1-jt(1-\ln(jt))}{j^2}+ O_t\left(\frac{\ln n}{n}\right).$$

The authors extend these results to $k>2,$ but I am mostly interested in the $k=2$ case.

My Question: What is the implied $t-$dependent constant in the $O_t(\ln n/n)$ term?

Answer

Attribution
Source : Link , Question Author : kodlu , Answer Author : Community