Universality of composite functions of periodic zeta functions

A. Laurinčikas

Sbornik. Mathematics, Tome 203 (2012) no. 11, pp. 1631-1646

Résumé

In the paper, we prove the universality, in the sense of Voronin, for some classes of composite functions $F(\zeta(s;\mathfrak a))$, where the function $\zeta(s;\mathfrak a)$ is defined by a Dirichlet series with periodic multiplicative coefficients. We also study the universality of functions of the form $F(\zeta(s;\mathfrak a_1),\dots,\zeta(s;\mathfrak a_r))$. For example, it follows from general theorems that every linear combination of derivatives of the function $\zeta(s;\mathfrak a)$ and every linear combination of the functions $\zeta(s;\mathfrak a_1),\dots,\zeta(s;\mathfrak a_r)$ are universal. Bibliography: 18 titles.

Détail

Keywords: support of a measure, periodic zeta function, limit theorem, the space of analytic functions, universality.

Bibliographie

[1] S. M. Voronin, “Theorem on the “universality” of the Riemann zeta-function”, Math. USSR-Izv., 9:3 (1975), 443–453 | DOI | MR | Zbl | Zbl

[2] S. M. Voronin, “O funktsionalnoi nezavisimosti Dirikhle $L$-funktsii”, Acta Arith., 27 (1975), 493–503 | MR | Zbl

[3] A. A. Karatsuba, S. M. Voronin, The Riemann zeta-function, de Gruyter Exp. Math., 5, de Gruyter, Berlin, 1992 | MR | MR | Zbl | Zbl

[4] A. Laurinčikas, Limit theorems for the Riemann zeta-function, Math. Appl., 352, Kluwer Acad. Publ., Dordrecht, 1996 | MR | Zbl

[5] A. Laurinčikas, R. Garunkštis, The Lerch zeta-function, Kluwer Acad. Publ., Dordrecht, 2002 | MR | Zbl

[6] J. Steuding, Value-distribution of $L$-functions, Lecture Notes in Math., 1877, Spinger-Verlag, Berlin–Heidelberg–New York, 2007 | DOI | MR | Zbl

[7] A. Laurinčikas, “The universality of zeta-functions”, Acta Appl. Math., 78:1–3 (2003), 251–271 | DOI | MR | Zbl

[8] K. Matsumoto, “Probabilistic value-distribution theory of zeta-functions”, Sugaku Expositions, 17:1 (2004), 51–71 | MR | Zbl

[9] K. Matsumoto, “An introduction to the value-distribution theory of zeta-functions”, Šiauliai Math. Semin., 1:9 (2006), 61–83 | MR | Zbl

[10] B. Bagchi, The statistical behaviour and universality properties of the Riemann zeta-function and allied Dirichlet series, Thesis, Indian Stat. Inst., Calcutta, 1981

[11] S. M. Gonek, Analytic properties of zeta and $L$-functions, Ph. D. Thesis, University of Michigan, 1979 | MR

[12] V. Garbaliauskienė, The universality of $L$-functions of elliptic curves, Vilnius University, Vilnius, 2005

[13] A. Laurinčikas, D. Šiaučiūnas, “Remarks on the universality of the periodic zeta function”, Math. Notes, 80:4 (2006), 532–538 | DOI | MR | Zbl

[14] J. Kaczorowski, “Some remarks on the universality of periodic $L$-functions”, New directions in value-distribution theory of zeta and $L$-functions (Würzburg, Germany, 2008), Ber. Math., Shaker Verlag, Aachen, 2008, 113–120 | MR | Zbl

[15] A. Laurinčikas, R. Macaitiene, “On the joint universality of periodic zeta functions”, Math. Notes, 85:1 (2009), 51–60 | DOI | MR | Zbl

[16] P. Billingsley, Convergence of probability measures, Wiley, New York–London–Sydney, 1968 | MR | MR | Zbl

[17] S. N. Mergelyan, “Uniform approximations to functions of a complex variable”, Amer. Math. Soc., Transl., 101 (1954) | MR | Zbl | Zbl

[18] J. L. Walsh, Interpolation and approximation by rational functions in the complex domain, Amer. Math. Soc., Providence, RI, 1965 | MR | Zbl