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Logarithmic derivatives of Artin $L$-functions

Part of: Multiplicative number theory Zeta and $L$-functions: analytic theory Algebraic number theory: global fields

Published online by Cambridge University Press:  26 February 2013

Peter J. Cho  and
Henry H. Kim
Peter J. Cho
Affiliation:
Department of Mathematics, University of Toronto, Toronto, Ontario, M5S 2E4, Canada (email: jcho@math.toronto.edu)
Henry H. Kim
Affiliation:
Department of Mathematics, University of Toronto, Toronto, Ontario, M5S 2E4, Canada Korea Institute for Advanced Study, Seoul, South Korea (email: henrykim@math.toronto.edu)
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Abstract

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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 $A_5$.

Keywords

strong Artin conjectureArtin ${L}$-functionlogarithmic derivatives of ${L}$-functionsEuler–Kronecker constants

MSC classification

Secondary: 11R42: Zeta functions and $L$-functions of number fields 11M41: Other Dirichlet series and zeta functions 11N75: Applications of automorphic functions and forms to multiplicative problems
Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 4 , April 2013 , pp. 568 - 586
Copyright
Copyright © 2013 The Author(s)

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