Skip to main content
SpringerLink
Log in

On the value distribution of L-functions of the Selberg class

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

In this paper, two limit theorems in the sense of the weak convergence of probability measures in the space of meromorphic functions and on the complex plane are proved for L-functions from a subclass of the Selberg class. The explicit form of the limit measures is given.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. P. Billingsley, Convergence of Probability Measures, Wiley, New York (1968).

    MATH  Google Scholar 

  2. H. Bohr and B. Jessen, “Über die Wertverteilung der Riemannschen Zetafunktion,” Acta Math., 54, 1–35 (1930).

    Article  MATH  MathSciNet  Google Scholar 

  3. H. Bohr and B. Jessen, “Über die Wertverteilung der Riemannschen Zetafunktion. II,” Acta Math., 58, 1–55 (1932).

    Article  MathSciNet  Google Scholar 

  4. E. Bombieri and D. A. Hejhal, “On the distribution of zeros of linear combinations of Euler products,” Duke Math. J., 80, 821–862 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  5. E. Bombieri and A. Perelli, “Distinct zeros of L-functions,” Acta Arith., 83, 271–281 (1998).

    MATH  MathSciNet  Google Scholar 

  6. J. B. Conrey and A. Ghosh, “On the Selberg class of Dirichlet series: small degrees,” Duke Math. J., 72, 673–693 (1993).

    Article  MATH  MathSciNet  Google Scholar 

  7. J. B. Conrey and A. Ghosh, Remarks on the Generalized Lindelöf Hypothesis, Preprint, www.math.obstate.edu/~conrey/papers.html.

  8. H. Cramér and M. R. Leddbetter, Stationary and Related Stochastic Processes, Wiley, New York (1967).

    MATH  Google Scholar 

  9. J. Genys and A. Laurinčikas, “A joint limit theorem for general Dirichlet series,” Lith. Math. J., 44, No. 1, 18–35 (2004).

    Article  MATH  Google Scholar 

  10. H. Heyer, Probability Measures on Locally Compact Groups, Springer, Berlin (1977).

    MATH  Google Scholar 

  11. D. Joyner, Distribution Theorems of L-Functions, Longman Scientific, Harlow (1986).

    MATH  Google Scholar 

  12. J. Kaczorowski and A. Perelli, “The Selberg class: a survey,” in: Number Theory in Progress. Proc. of the Int. Conf. in Honor of the 60th Birthday of A. Schinzel (Zakopane, 1997), Vol. 2, Elementary and Analytic Number Theory, De Gruyter, Berlin (1999), pp. 953–992.

    Google Scholar 

  13. A. Laurinčikas, Limit Theorems for the Riemann Zeta-Function, Kluwer, Dordrecht (1996).

    Google Scholar 

  14. A. Laurinčikas, “Application of probabilistic methods in the theory of the Riemann zeta-function,” in: Int. Conf. Modern Problems of Number Theory and Its Applications (Russia, Tula, 2001 ). Topical problems. Pt. II, Moscow State University, Moscow (2002), pp. 98–116.

    Google Scholar 

  15. A. Laurinčikas and R. Garunkštis, The Lerch Zeta-Function, Kluwer, Dordrecht (2002).

    MATH  Google Scholar 

  16. A. Laurinčikas and K. Matsumoto, “Joint value-distribution on Lerch zeta-functions,” Lith. Math. J., 38, No. 3, 238–249 (1998).

    Article  MATH  Google Scholar 

  17. R. Macaitienė, “Joint discrete limit theorems in the space of analytic functions for general Dirichlet series,” Lith. Math. J., 45, No. 1, 84–93 (2005).

    Article  MATH  Google Scholar 

  18. K. Matsumoto, “Probabilistic value-distribution theory of zeta-functions,” Sugaku Expositions, 17, No. 1, 51–71 (2004).

    MATH  MathSciNet  Google Scholar 

  19. G. Molteni, “A note on a result of Bochner and Conrey–Ghosh about the Selberg class,” Arch. Math., 72, 219–222 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  20. A. Perelli, “General L-functions,” Ann. Mat. Pura Appl., 130, 287–306 (1982).

    Article  MATH  MathSciNet  Google Scholar 

  21. A. Perelli, “A survey of the Selberg class of L-functions. I,” Milan J. Math., 73, 19–52 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  22. A. Perelli, “A survey of the Selberg class of L-functions. II,” Riv. Mat. Univ. Parma, 7, No. 3, 83–118 (2004).

    MathSciNet  Google Scholar 

  23. A. Selberg, “Old and new conjectures and results about a class of Dirichlet series,” in: E. Bombieri et al., eds., Proc. of the Amalfi Conf. on Analytic Number Theory (Maiori, 1989 ), Univ. Salerno, Salerno (1992), pp. 367–385.

    Google Scholar 

  24. J. Steuding, “On the universality for functions in the Selberg class,” in: D. R. Heath–Brown and B. Z. Moroz, eds., Proc. of the Session in Analytic Number Theory and Diophantine Equations (Bonn, 2002), Bonner Math. Schriften, Vol. 360, Bonn (2003).

  25. J. Steuding, Value-distribution of L-functions and allied zeta-functions — with an emphasis on aspects of universality, Habilitationschrift, J. W. Goethe-Universität, Frankfurt (2003).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Siauliai University, Šiauliai State College, Šiauliai, Lithuania

    R. Macaitienė

Authors
  1. R. Macaitienė
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to R. Macaitienė.

Additional information

Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 16, No. 5, pp. 103–116, 2010.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Macaitienė, R. On the value distribution of L-functions of the Selberg class. J Math Sci 180, 599–609 (2012). https://doi.org/10.1007/s10958-012-0659-9

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-012-0659-9

Keywords

Search

Navigation