Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-10T17:48:07.110Z Has data issue: false hasContentIssue false
  • English
  • Français

Logarithmic derivatives of Artin $L$-functions

Part of: Zeta and $L$-functions: analytic theory Multiplicative number 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)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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)

References

[Cho]Cho, P. J., The strong Artin conjecture and number fields with large class numbers, Quart. J. Math., to appear.Google Scholar
[CKa]Cho, P. J. and Kim, H. H., Dihedral and cyclic extensions with large class numbers, J. Théor. Nombres Bordeaux, to appear.Google Scholar
[CKb]Cho, P. J. and Kim, H. H., Application of the strong Artin conjecture to class number problem, Canad. J. Math., to appear.Google Scholar
[CKc]Cho, P. J. and Kim, H. H., Weil’s theorem on rational points over finite fields and Artin $L$-functions, Conference Proceedings of Tata Institute, to appear.Google Scholar
[Coh93]Cohen, H., A course in computational algebraic number theory (Springer, Berlin, 1993).Google Scholar
[Dai06]Daileda, R. C., Non-abelian number fields with very large class numbers, Acta Arith. 125 (2006), 215–255.Google Scholar
[Duk03]Duke, W., Extreme values of Artin L-functions and class numbers, Compositio Math. 136 (2003), 103–115.CrossRefGoogle Scholar
[Duk04]Duke, W., Number fields with large class groups, in Number theory, CRM Proceedings and Lecture Notes, vol. 36 (American Mathematical Society, Providence, RI, 2004), 117–126.Google Scholar
[Iha06]Ihara, Y., On the Euler–Kronecker constants of global fields and primes with small norms, in Algebraic geometry and number theory, Progress in Mathematics, vol. 253 (Birkhäuser, Boston, MA, 2006), 407–451.Google Scholar
[IMS09]Ihara, Y., Murty, V. K. and Shimura, M., On the logarithmic derivatives of Dirichlet $L$-functions at $s=1$, Acta Arith. 137 (2009), 253–276.Google Scholar
[IK04]Iwaniec, H. and Kowalski, E., Analytic number theory, American Mathematical Society Colloquium Publications, vol. 53 (American Mathematical Society, Providence, RI, 2004).Google Scholar
[KW09]Khare, C. and Wintenberger, J. P., Serre’s modularity conjecture. I, Invent. Math. 178 (2009), 485–504.Google Scholar
[Kim04]Kim, H. H., An example of non-normal quintic automorphic induction and modularity of symmetric powers of cusp forms of icosahedral type, Invent. Math. 156 (2004), 495–502.Google Scholar
[Kli98]Klingen, N., Arithmetical similarities, Oxford Mathematical Monographs (Oxford University Press, New York, 1998).Google Scholar
[KM02]Kowalski, E. and Michel, P., Zeros of families of automorphic $L$-functions close to $1$, Pacific J. Math. 207 (2002), 411–431.CrossRefGoogle Scholar
[MM]Mourtada, M. and Murty, V. K., An omega theorem for $\frac {L'}{L}(1,\chi _D)$, Int. J. Number Theory, to appear.Google Scholar
[Nai76]Nair, M., Power free values of polynomials, Mathematika 23 (1976), 159–183.Google Scholar
[Ram00]Ramakrishnan, D., Modularity of the Rankin-Selberg $L$-series, and multiplicity one for ${\rm SL}(2)$, Ann. of Math. 152 (2000), 45–111.Google Scholar
[Roj03]Rojas, M., Dedekind zeta functions and the complexity of Hilbert’s Nullstellensatz, Preprint (2003), arXiv:math/0301111.Google Scholar
[Sch04]Schmidt, W., Equations over finite fields: an elementary approach (Kendrick Press, Heber City, UT, 2004).Google Scholar
[Ser03]Serre, J. P., On a theorem of Jordan, Bull. Amer. Math. Soc. (N.S.) 40 (2003), 429–440.Google Scholar
[Ser08]Serre, J. P., Topics in Galois theory, Research Notes in Mathematics (A K Peters, Wellesley, MA, 2008).Google Scholar
[Smi00]Smith, G. W., Polynomials over $\Bbb Q(t)$ and their Galois groups, Math. Comp. 230 (2000), 775–796.Google Scholar