We may also read it as follows: $$\frac{\frac{c}{d}\wedge\frac{m}{n}}{\frac{c}{d}\vee\frac{m}{n}} = \frac{cn\wedge dm}{cn\vee dm} $$

No. Do you know an integral formula for the generalized harmonic number?

Also, are these regularization terms necessary if we impose $\Im(s)\neq 0$?

Very instructive comment. Could you elaborate a little bit more on the meaning of the regularization terms for $0<|\Re(s)|<1$ and how to get them?

I would be glad to have some insight on how to get such asymptotic formula. I think we should have at least the absolute convergence of the series for $\Im s =0$ and $\Re s >1$ $\forall t\in \mathbb{R}$, and hopefully the convergence for $\Im s = 0$ and $\forall \Re s$ for $t\neq 0$. Though I am not sure how to prove this.

Thank you for your answer. Your $f(s)$ was indeed my starting point, but I think I was not clear enough, I will add precisions. Actually, I am assuming $s,t\in \mathbb{R}$ and I would like to study the convergence of the Dirichlet series that yields $f(s,t)$ or the asymptotics of $A_n(t)$ for non zero $t$ (to avoid the pole at $s=1$).