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Asymptotic behaviour of the Euler-Kronecker constant

  • M. A. Tsfasman
Part of the Progress in Mathematics book series (PM, volume 253)

Summary

This appendix to the beautiful paper [1] of Ihara puts it in the context of infinite global fields of our papers [2] and [3]. We study the behaviour of Euler-Kronecker constant γK when the discriminant (genus in the function field case) tends to infinity. Results of [2] easily give us good lower bounds on the ratio \( {{\gamma _K } \mathord{\left/ {\vphantom {{\gamma _K } {\log \sqrt {\left| {d_K } \right|} }}} \right. \kern-\nulldelimiterspace} {\log \sqrt {\left| {d_K } \right|} }} \). In particular, for number fields, under the generalized Riemann hypothesis we prove
$$ \lim \inf \frac{{\gamma _K }} {{\log \sqrt {\left| {d_K } \right|} }} \geqslant - 0.26049.... $$
Then we produce examples of class-field towers, showing that
$$ \lim \inf \frac{{\gamma _K }} {{\log \sqrt {\left| {d_K } \right|} }} \leqslant - 0.17849.... $$

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References

  1. [1]
    Y. Ihara, On the Euler-Kronecker constants of global fields and primes with small norms, in V. Ginzburg, ed., Algebraic Geometry and Number Theory: In Honor of Vladimir Drinfeld’s 50th Birthday, Progress in Mathematics, Vol. 850, Birkhäuser Boston, Cambridge, MA, 2006, 407–452 (this volume).Google Scholar
  2. [2]
    M. A. Tsfasman and S. G. Vlăduţ, Infinite global fields and the generalized Brauer-Siegel theorem, Moscow Math. J., 2-2 (2002), 329–402.zbMATHGoogle Scholar
  3. [3]
    M. A. Tsfasman and S. G. Vlăduţ, Asymptotic properties of zeta-functions, J. Math. Sci. (N.Y.), 84-5 (1997), 1445–1467.zbMATHCrossRefGoogle Scholar
  4. [4]
    F. Hajir and C. Maire, Tamely ramified towers and discriminant bounds for number fields II, J. Symbol. Comput., 33-4 (2002), 415–423.zbMATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    A. Zykin, Private communication.Google Scholar

Copyright information

© Birkhäuser Boston 2006

Authors and Affiliations

  • M. A. Tsfasman
    • 1
    • 2
  1. 1.Poncelet LaboratoryUMI 2615 of CNRS and the Independent University of MoscowMoscowRussia
  2. 2.Institut de Mathématiques de LuminyMarseilleFrance

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