Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-22T18:19:07.185Z Has data issue: false hasContentIssue false
  • English
  • Français

The joint distribution of the Riemann zeta - function

Published online by Cambridge University Press:  17 April 2009

A. Laurinčikas
A. Laurinčikas
Affiliation:
Department of Mathematics and Informatics, Vilnius University, Naugarduko, 24, 03225 Vilnius, Lithuania e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Extract

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.

In the paper the asymptotic distribution of (|ζ(s)|,ζ(s)), where ζ(s) is the Riemann zeta - function, in the sense of weak convergence of probability measures is considered. For this, the continuity theorems for probability measures on ℝ × ℂ are used. Some aspects of the dependence of |ζ(s)| and ζ(s) are also discussed.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 73 , Issue 2 , April 2006 , pp. 201 - 217
Copyright
Copyright © Australian Mathematical Society 2006

References

[1]Bagchi, B., The statistical behavior and universality properties of the Riemann zeta-function and other allied Dirichlet series, (Ph.D. Thesis) (Indian Statistical Institute, Calcutta, 1981).Google Scholar
[2]Belov, I., ‘A limit theorem for the Dirichlet L-functions on the complex plane’, Chebyshevskiǰ Sb. 4 (2003), 122–133.Google Scholar
[3]Billingsley, P., Convergence of probability measures (J. Wiley and Sons, New York, 1962).Google Scholar
[4]Bohr, H. and Jessen, B., ‘Über die Wertverteilung der Riemannschen Zeta funktion’, Erste Mitteilung, Acta Math. 541 (930), 1–35.Google Scholar
[5]Bohr, H. and Jessen, B., ‘Über die Wertverteilung der Riemannschen Zeta funktion’, Zweite Mitteilung, Acta Math. 58 (1932), 1–55.Google Scholar
[6]Joyner, D., Distribution Theorems of L-functions, Pitman Res. Notes in Math. Ser. 142 (Longman Sci. & Tech., Harlow, 1986).Google Scholar
[7]Heyer, H., Probability nmeasures on locally compact groups (Springer-Verlag, Berlin, 1977).Google Scholar
[8]Laurinčikas, A., ‘Distribution of values of complex-valued functions’, (in Russian), Litovsk. Mat. Sb. 15 (1975), 123–134.Google Scholar
[9]Laurinčikas, A., Limit theorems for the Riemann zeta-function (Kluwer Academic Publishers, Dordrecht, 1996).CrossRefGoogle Scholar
[10]Laurinčikas, A., ‘Limit theorems for the Matsumoto zeta-function’, J. Théor. Nombres Bordeaux 8 (1996), 143–158.CrossRefGoogle Scholar
[11]Laurinčikas, A., ‘On limit distribution of the Matsumoto zeta-function’, Lithuanian Math. J. 36 (1996), 371–387.Google Scholar
[12]Laurinčikas, A., ‘On limit distribution of the Matsumoto zeta-function’, Acta Arith. 79 (1997), 31–39.CrossRefGoogle Scholar
[13]Matsumoto, K., ‘Value - distribution of zeta-functions’, in Analytic Number Theory, (Nagasaka, K. and Fouvry, E., Editors), Lecture Notes in Math. 1437, 1990, pp. 178–187.CrossRefGoogle Scholar
[14]Matsumoto, K., ‘Probabilistic value - distribution theory of zeta-functions’, Sūgaku Expositions 17 (2004), 51–71.Google Scholar