The Chebyshev equivalent is well-known: $$\sum_{p < x} \frac{\log p}{p} \sim \log x.$$

I am interested in the number field setting. How can I deduce (for instance I tried to encapsulated ideals of constant norm) a similar equivalent in for prime ideals, i.e. is there a constant $C$ such that $$\sum_{n(\mathfrak{p}) < x} \frac{\log N\mathfrak{p}}{N\mathfrak{p}} \sim C \log x,$$

where the sum runs over prime ideals and $n$ is the norm on the number field? The constant should probably be a residue of the Dedekind zeta function on the field. Is it straightforward? Or do you have any reference?


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