Logarithmic derivatives of Artin L-functions

Abstract

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 vertical bar d(K)vertical bar and (n 1) log log vertical bar d(K)vertical bar, respectively. Unconditionally, we show that there are in finitely 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