Asymptotic properties of Dedekind zeta functions in families of number fields

Alexey Zykin[1]

  • [1] State University — Higher School of Economics, 7, Vavilova st. 117312, Moscow, Russia

Journal de Théorie des Nombres de Bordeaux (2010)

  • Volume: 22, Issue: 3, page 771-778
  • ISSN: 1246-7405

Abstract

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The main goal of this paper is to prove a formula that expresses the limit behaviour of Dedekind zeta functions for ℜ s > 1 / 2 in families of number fields, assuming that the Generalized Riemann Hypothesis holds. This result can be viewed as a generalization of the Brauer–Siegel theorem. As an application we obtain a limit formula for Euler–Kronecker constants in families of number fields.

How to cite

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Zykin, Alexey. "Asymptotic properties of Dedekind zeta functions in families of number fields." Journal de Théorie des Nombres de Bordeaux 22.3 (2010): 771-778. <http://eudml.org/doc/116434>.

@article{Zykin2010,
abstract = {The main goal of this paper is to prove a formula that expresses the limit behaviour of Dedekind zeta functions for $\Re s &gt; 1/2$ in families of number fields, assuming that the Generalized Riemann Hypothesis holds. This result can be viewed as a generalization of the Brauer–Siegel theorem. As an application we obtain a limit formula for Euler–Kronecker constants in families of number fields.},
affiliation = {State University — Higher School of Economics, 7, Vavilova st. 117312, Moscow, Russia},
author = {Zykin, Alexey},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Dedekind zeta function; asymptotically exact families},
language = {eng},
number = {3},
pages = {771-778},
publisher = {Université Bordeaux 1},
title = {Asymptotic properties of Dedekind zeta functions in families of number fields},
url = {http://eudml.org/doc/116434},
volume = {22},
year = {2010},
}

TY - JOUR
AU - Zykin, Alexey
TI - Asymptotic properties of Dedekind zeta functions in families of number fields
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2010
PB - Université Bordeaux 1
VL - 22
IS - 3
SP - 771
EP - 778
AB - The main goal of this paper is to prove a formula that expresses the limit behaviour of Dedekind zeta functions for $\Re s &gt; 1/2$ in families of number fields, assuming that the Generalized Riemann Hypothesis holds. This result can be viewed as a generalization of the Brauer–Siegel theorem. As an application we obtain a limit formula for Euler–Kronecker constants in families of number fields.
LA - eng
KW - Dedekind zeta function; asymptotically exact families
UR - http://eudml.org/doc/116434
ER -

References

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  2. Y. Ihara, On the Euler–Kronecker constants of global fields and primes with small norms. Algebraic geometry and number theory, 407–451, Progr. Math., 253, Birkhaüser Boston, Boston, MA, 2006. Zbl1185.11069MR2263195
  3. H. Iwaniec, E. Kowalski, Analytic number theory. American Mathematical Society Colloquium Publications, 53. AMS, Providence, RI, 2004. Zbl1059.11001MR2061214
  4. H. Iwaniec, W. Luo, P. Sarnak, Low lying zeros of families of L -functions. Inst. Hautes Études Sci. Publ. Math., Num. 91 (2000), 55–131. Zbl1012.11041MR1828743
  5. H. Iwaniec, P. Sarnak, Dirichlet L -functions at the central point. Number theory in progress, Vol. 2 (Zakopane-Koscielisko, 1997), 941–952, de Gruyter, Berlin, 1999. Zbl0929.11025MR1689553
  6. H. Iwaniec, P. Sarnak, The nonvanishing of central values of automorphic L -functions and Siegel’s zero. Israel J. Math. A 120 (2000), 155–177. Zbl0992.11037MR1815374
  7. S. Lang, Algebraic number theory. 2nd ed. Graduate Texts in Mathematics 110, Springer-Verlag, New York, 1994. Zbl0811.11001MR1282723
  8. H. M. Stark, Some effective cases of the Brauer-Siegel Theorem. Invent. Math. 23(1974), pp. 135–152. Zbl0278.12005MR342472
  9. E. C. Titchmarsh, The theory of functions. 2nd ed. London: Oxford University Press. X, 1975. Zbl0336.30001MR197687
  10. M. A. Tsfasman, Asymptotic behaviour of the Euler-Kronecker constant. Algebraic geometry and number theory, 453–458, Progr. Math., 253, Birkhaüser Boston, Boston, MA, 2006. Zbl1185.11070MR2263196
  11. M. A. Tsfasman, S. G. Vlăduţ, Infinite global fields and the generalized Brauer–Siegel Theorem. Moscow Mathematical Journal, Vol. 2(2002), Num. 2, 329–402. Zbl1004.11037MR1944510
  12. A. Zykin, Brauer–Siegel and Tsfasman–Vlăduţ theorems for almost normal extensions of global fields. Moscow Mathematical Journal, Vol. 5 (2005), Num 4, 961–968. Zbl1125.11062MR2267316
  13. A. Zykin, Asymptotic properties of zeta functions over finite fields. Preprint. Zbl1329.14056

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