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In a family of $S_{d+1}$-fields ($d=2,3,4$), we obtain the conjectured upper and
lower bounds of the residues of Dedekind zeta functions except for a density zero set.
For $S_5$-fields, we need to assume the strong Artin conjecture.
We also show that there exists an infinite family of number fields with the upper and lower bounds, resp.
This is a joint work with Henry Kim