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Let $K$ be a number field of degree $n$ , and let $d_K$ be its discriminant. Then, under the Artin conjecture, the generalized Riemann hypothesis and a certain zero-density hypothesis, we show that the upper and lower bounds of the logarithmic derivatives of Artin $L$ -functions attached to $K$ at $s=1$ are $\log \log |d_K|$ and $-(n-1) \log \log |d_K|$ , respectively. Unconditionally, we show that there are infinitely many number fields with the extreme logarithmic derivatives; they are families of number fields whose Galois closures have the Galois group $C_n$ for $n=2,3,4,6$ , $D_n$ for $n=3,4,5$ , $S_4$ or $A_5$ .

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Logarithmic derivatives of Artin $L$ -functions

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Logarithmic derivatives of Artin LL-functions

Logarithmic derivatives of Artin $L$ -functions