## Abstract

Here we prove a limit theorem in the sense of the weak convergence of probability measures in the space of meromorphic functions for a general Dirichlet series. The explicit form of the limit measure in this theorem is given.

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### formerly Central European Journal of Mathematics

# Discrete limit theorems for general Dirichlet series. III

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*Analysis*, 2007, Volume 26, Number 3*Acta Applicandae Mathematicae*, 2007, Volume 97, Number 1-3, Page 99*Analysis*, 2007, Volume 26, Number 3, Page 373

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

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A. Laurinčikas / R. Macaitienė

Here we prove a limit theorem in the sense of the weak convergence of probability measures in the space of meromorphic functions for a general Dirichlet series. The explicit form of the limit measure in this theorem is given.

Keywords: Dirichlet series; probability measure; random element; weak convergence

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[2] H. Bohr and B. Jessen: “Über die Werverteilung der Riemannschen Zeta function”. Erste Mitteilung.Acta Math., Vol. 54, (1930), pp. 1–35. [Crossref]

[3] H. Bohr and B. Jessen: “Über die Werverteilung der Riemannschen Zeta function”. Zweite Mitteilung,Acta Math., Vol. 54, (1932), pp. 1–55.

[4] J. Genys and A. Laurinčikas: “Value distribution of general Dirichlet series IV”,Lith. Math. J., Vol. 43, No. 3, (2003), pp. 342–358;Lith. Math. J., Vol. 43, No. 3, (2003), pp. 281–294 (in Russian). http://dx.doi.org/10.1023/A:1026189318741 [Crossref]

[5] A. Laurinčikas:Limit Theorems for the Riemann Zeta-Function, Kluwer, Dordrecht, 1996.

[6] A. Laurinčikas: “Value distribution of general Dirichlet series”, In: B. Grigelionis et al. (Eds.):Probab. Theory and Math. Statistics; Proceedings of the seventh Vilnius, TEV, Vilnius, (1999), pp. 405–414.

[7] A. Laurinčikas: “Value distribution of general Dirichlet series. II”,Lith. Math. J., Vol. 41, No. 4, (2001), pp. 351–360. http://dx.doi.org/10.1023/A:1013860521038 [Crossref]

[8] A. Laurinčikas: “Limit theorems for general Dirichlet series”,Theory Stoch. Proc., Vol. 8, No. 24, (2002), pp. 256–268.

[9] A. Laurinčikas, W. Schwarz and J. Steuding: “Value distribution of general Dirichelet series. III”, In: A. Dubickas et al. (Eds.):Analytic and Probab. Methods in Number Theory. Proc. The Third Palanga Conf., TEV, Vilnius, (2002), pp. 137–156.

[10] A. Laurinčikas and R. Macaitienė: “Discrete limit theorems for general Dirichlet series. I,”Chebyshevski sbornik, Vol. 4, No. 3, (2003), pp. 156–170.

[11] R. Macaitienė: “Discrete limit theorems for general Dirichlet polynomials”,Lith. Math. J., Vol. 42 (spec. issue), (2002), pp. 705–709.

[12] R. Macaitienė: “Discrete limit theorems for general Dirichlet series. II”,Lith. Math. J., (to appear).

[13] H.L. Montgomery:Topics in multiplicative number theory, Springer, Berlin, 1971.

[14] E.M. Nikishin: “Dirichlet series with independent exponents and certain of their applications”,Matem.sb, Vol. 96, No. 1, (1975), pp. 3–40 (in Russian).

[15] Y.G. Sinai:Introduction to Ergodic Theory, Princeton Univ. Press, 1976.

[16] A.A. Tempelman:Ergodic Theorems on Groups, Mokslas, Vilnius, 1986.

**Published Online**: 2004-06-01

**Published in Print**: 2004-06-01

**Citation Information: **Open Mathematics, ISSN (Online) 2391-5455, DOI: https://doi.org/10.2478/BF02475231. Export Citation

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[1]

Renata Macaitiene

[2]

Renata Macaitienė

[3]

Renata Macaitiene

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