Let $K$ be a number field, we denote $\zeta_K(s)$ by the corresponding Dedekind zeta function. For a complex number $s$, we denote $\sigma, \tau$ by its real and imaginary parts respectively.

Question 1. Can we find positive constants $c,d$ such that for $|\tau|\geq d$ and $\sigma\geq 1-c/ \log |\tau|$, we have

  1. $\zeta_K'(s)/\zeta_K(s)\ll \log|\tau|$;
  2. $1/\zeta_K(s)\ll \log |\tau|$;
  3. $|\log\zeta_K(s)|\ll \log\log|\tau|+O(1)$.

When $K=\mathbb{Q}$, the above assertion is classical, cf. Theorem 3.22 of Tenenbaum's book. I guess similar inequalities hold for all Dedekind zeta functions, and the proof should follow the similar methods, but I am not able to find a desired theorem or proof in the literature.

So could you please give me a reference? Thanks a lot!


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