MODIFICATION OF THE MISHOU THEOREM
https://doi.org/10.22405/2226-8383-2016-17-3-135-147
Abstract
The Mishou theorem asserts that a pair of analytic functions from a wide class can be approximated by shifts of the Riemann zeta and Hurwitz zeta-functions \((\zeta(s+i\tau), \zeta(s+i\tau, \alpha))\) with transcendental \(\alpha\), \(\tau\in\mathbb{R}\), and that the set of such \(\tau\) has a positive lower density. In the paper, we prove that the above set has a positive density for all but at most countably many \(\varepsilon>0\), where \(\varepsilon\) is the accuracy of approximation. We also obtain similar results for composite functions \(F(\zeta(s),\zeta(s,\alpha))\) for some classes of operator \(F\).
About the Authors
A. Laurinčikas
Vilnius University
Russian Federation
Full member of the AS in Lithuania, doctor of physical and mathematical sciences, professor, head of probability theory’s and number theory’s chair
Russian Federation
Full member of the AS in Lithuania, doctor of physical and mathematical sciences, professor, head of probability theory’s and number theory’s chair
L. Meška
Vilnius University
Russian Federation
doctoral student of the Faculty of Mathematics and Informatics
Russian Federation
doctoral student of the Faculty of Mathematics and Informatics
References
1. Billingsley P. Convergence of Probability Measures. New York: Willey, 1968.
2. Javtokas A., Laurinˇcikas A. Universality of the periodic Hurwitz zeta-function // Integr. Transf. Spec. Funct. 2006. Vol. 17. P. 711–722.
3. Kaˇcinskait˙e R., Laurinˇcikas A. The joint distribution of periodic zeta-functions // Studia Sci. Math. Hung. 2011. Vol. 18. P. 257–279.
4. Лауринчикас А.П. Аналог теоремы Воронина для периодических дзета-функций Гурвица // Матем. Сб. 2007. Т. 198, №. 2. С. 91–102.
5. Лауринчикас А. Совместная универсальность периодических дзета-функций Гурвица // Изв. РАН. Сер. матем. 2008. Т. 72, №. 4. С. 121–140.
6. Laurinˇcikas A. The joint universality of Hurwitz zeta-functions // Siauliai Math. Semin. 2008. ˇ Vol. 3(11). P. 169–187.
7. Лауринчикас А. Совместная универсальность дзета- функций с периодическими коэффициентами // Изв. РАН. Сер. матем. 2010. Т. 74, №. 3. С. 79–102.
8. Laurinˇcikas A. Universality of composite functions // Functions in Number Theory and Their Probabilistic Aspects, K. Matsumoto et al (Eds), RIMS Kˆokyˆuroku Bessatsu. 2012. Vol. B34. P. 191–204.
9. Laurinˇcikas A. On joint universality of the Riemann zeta-function and Hurwitz zeta-functions // J. Number Theory. 2012. Vol. 132. P. 2842–2853.
10. Лауринчикас А. Расширение универсальности дзета функций с периодическими коэффициентами // Сиб. матем. ж. 2016. Т. 57, №. 2. С. 420–431.
11. Laurinˇcikas A., Garunkˇstis R. The Lerch Zeta-Function. Dordrecht: Kluwer, 2002.
12. Laurinˇcikas A., Matsumoto K. The universality of zeta-functions attached to certain cusp forms // Acta Arith. 2001. Vol. 98. P. 345–359.
13. Лауринчикас А., Мешка Л. Уточнение неравенства универсальности // Матем. заметки. 2014. Т. 96, №. 6. С. 905–910.
14. Laurinˇcikas A., Meˇska L. On the modification of the universality of the Hurwitz zeta-function // Nonlinear Analysis: Modelling and Control. 2016. Vol. 21, No. 4. P. 564–576.
15. Лауринчикас А.П., Шяучюнас Д. Замечания об универсальности периодической дзета-
16. функции // Матем. заметки. 2006. Т. 80, №. 4. С. 561–568.
17. Matsumoto K. A survey on the theory of universality for zeta and
18. Мергелян С.Н. Равномерные приближения функций комплексного переменного // УМН. 1952. Т. 7, №. 2. С. 31–122
19. Meˇska L. A modification of the universality inequality // Siauliai Math. Semin. 2014. Vol. 9(17). ˇ P. 71–81.
20. Mishou H. The joint value distribution of the Riemann zeta-function and Hurwitz zeta-functions // Lith. Math. J. 2007. Vol. 47. P. 32–47.
21. Steuding J. Value-Distribution of
22. Воронин С. М. Теорема об “универсальности” дзета-функции Римана // Изв. АН СССР. Сер. матем. 1975. Т. 39. С. 475–486.
Review
For citations:
Laurinčikas A., Meška L. MODIFICATION OF THE MISHOU THEOREM. Chebyshevskii Sbornik. 2016;17(3):135-147. (In Russ.) https://doi.org/10.22405/2226-8383-2016-17-3-135-147
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