There are many 'effective' versions of asymptotic theorems. This paper describes an effective version of a theorem as 'an asymptotic with an explicit error term, which holds for x larger than an explicit quantity, in which any dependence [on variable quantities] is specified, and all constants can be computed.'
For example, the Chebotarev density theorem: Let $\pi_{\mathcal{C}}(x,L/k)=\#\{\mathfrak{p}\in \mathcal{O}_k: \mathfrak{p}$ unramified in $L, \left[\frac{L/k}{\mathfrak{p}}\right]=\mathcal{C}$, with norm less than or equal to $x\}$. Then Chebotarev's density theorem states that
$\pi_{\mathcal{C}}(x,L/k)\sim |\mathcal{C}|/|Gal(L/k)|Li(x)$.
Also, there are effective versions (the following one assuming GRH by Lagarias and Odlyzko):
$|\pi_{\mathcal{C}}(x,L/k)-|\mathcal{C}|/|Gal(L/k)|Li(x)|\leq C_0|\mathcal{C}|/|Gal(L/k)|x^{1/2}\log(D_Lx^{n_L})$, for $x$ above a certain explicit range and $C_0$ a computable constant.
Question: What fields / texts are available to learn how to turn an asymptotic estimate into an explicit 'effective' bound?