See Terence Tao's blog for the usual ones and how to derive them from Prime Number Theorem
https://terrytao.wordpress.com/2013/12/11/mertens-theorems/
I was particularly interested in the so-called third Mertens theorem, where $\gamma $ is Euler's constant:
\[ \prod _{0<p<x} \left( 1-\frac 1p \right) ^{-1} = e^{\gamma } \log x + O(1) . \]
I wanted to have an analogue for number fields $K$:
\[ \prod _{\mathfrak p ,\ N(\mathfrak p ) < x} \left( 1-\frac 1{N (\mathfrak p) } \right) ^{-1} = C_K \log x + O_K(1) \]
for some constant $C_K>0 $.
Such a Mertens theorem turned out to be already known. The paper by Rosen (link below) would suffice for me. The constant is
\[ C_K = e^{\gamma }\cdot \underset{s=1}{\operatorname{res}} \zeta _K(s) . \]
The proof consists of the Prime Ideal Theorem and Abel's summation method.
Michael Rosen 1999 (where is the published version?)
https://www.researchgate.net/publication/266272896_A_generalization_of_Mertens'_theorem
It's published in J. Ramanujan Math. Soc. 14 (1999), no. 1, 1--19, but there is no online version of this issue.
The following papers have more explicit error terms.
Lebacque 2007
Garcia, Lee 2022
https://link.springer.com/content/pdf/10.1007/s11139-021-00435-6.pdf
