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The joint universality and the functional independence for Lerch zeta-functions

Published online by Cambridge University Press:  22 January 2016

Antanas Laurinčikas  and
Kohji Matsumoto
Antanas Laurinčikas
Affiliation:
Department of Mathematics, Vilnius University, Naugarduko, 24, 2006 Vilnius, Lithuania, antanas.laurincikas@maf.vu.lt
Kohji Matsumoto
Affiliation:
Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya, 464-8602, Japan, kohjimat@math.nagoya-u.ac.jp
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Abstract

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The joint universality theorem for Lerch zeta-functions L(λl, αl, s) (1 ≤ l ≤ n) is proved, in the case when λls are rational numbers and αls are transcendental numbers. The case n = 1 was known before ([12]); the rationality of λls is used to establish the theorem for the “joint” case n ≥ 2. As a corollary, the joint functional independence for those functions is shown.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 157 , 2000 , pp. 211 - 227
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2000

References

[1] Amou, M. and Katsurada, M., Differential transcendence of a class of generalized Dirichlet series, in preparation.Google Scholar
[2] Bagchi, B., The statistical behaviour and universality properties of the Riemann zeta-function and other allied Dirichlet series, Ph. D. Thesis, Calcutta, Indian Statistical Institute, 1981.Google Scholar
[3] Bagchi, B., Joint universality theorem for Dirichlet L-functions, Math. Zeitschrift, 181 (1982), 319–334.Google Scholar
[4] Billingsley, P., Convergence of Probability Measures, John Wiley, New York, 1968.Google Scholar
[5] Garunkstis, R., The universality theorem with weight for the Lerch zeta-function, In: Analytic and Probabilistic Methods in Number Theory, New Trends in Probability and Statistics, Vol. 4, Laurincikas, A. et al. (Eds.), VSP/TEV, 1997, pp. 59–67.Google Scholar
[6] Garunkstis, R. and Laurincikas, A., On one Hilbert’s problem for the Lerch zeta-function, Publ. Inst. Math. (Beograd) (N.S.), 65 (79) (1999), 63–68.Google Scholar
[7] Gonek, S.M., Analytic properties of zeta and L-functions, Ph. D. Thesis, University of Michigan, 1979.Google Scholar
[8] Laurincikas, A., On the zeros of linear forms of Dirichlet series, Liet. Matem. Rink., 26(3) (1986), 468–477 (in Russian) = Lith. Math. J., 26(3) (1986), 244–251.Google Scholar
[9] Laurincikas, A., Application of probability methods in the theory of Dirichlet series, Thesis Doctor of Sciences, Vilnius, 1989 (in Russian).Google Scholar
[10] Laurincikas, A., Limit Theorems for the Riemann Zeta-Function, Kluwer Academic Publish ers, Dordrecht/ Boston/ London, 1996.Google Scholar
[11] Laurincikas, A., On limit distribution of the Lerch zeta-function, In: Analytic and Probabilis tic Methods in Number Theory, New Trends in Probability and Statistics, Vol.4, Laurincikas, A. et al.(Eds.), VSP/TEV, 1997, pp.135–148.Google Scholar
[12] Laurincikas, A., The universality of the Lerch zeta-function, Liet. Matem. Rink., 37(3) (1997), 367–375 (in Russian) = Lith. Math. J., 37(3) (1997), 275–280.Google Scholar
[13] Laurincikas, A., On the Lerch zeta-function with rational parameters, Liet. Matem. Rink., 38(1) (1998), 113–124 (in Russian) = Lith. Math. J., 38(1) (1998), 89–97.Google Scholar
[14] Laurincikas, A., On the zeros of linear combinations of the Matsumoto zeta-functions, Liet. Matem. Rink., 38(2) (1998), 185–204 (in Russian) = Lith. Math. J., 38(2) (1998), 144–159.Google Scholar
[15] Laurincikas, A. and Matsumoto, K., Joint value-distribution theorems on Lerch zeta-functions, Liet. Matem. Rink., 38(3) (1998), 312–326 = Lith. Math. J., 38(3) (1998), 238–249.Google Scholar
[16] Lerch, M., Note sur la fonction , Acta Math., 11 (1887), 19–24.Google Scholar
[17] Kuipers, L. and Niederreiter, H., Uniform Distribution of Sequences, John Wiley& Sons, New York/London/Sydney/Toronto, 1974.Google Scholar
[18] Voronin, S.M., On the functional independence of Dirichlet L-functions, Acta Arith., 27 (1975), 493–503 (in Russian).Google Scholar
[19] Voronin, S.M., Analytic properties of generating Dirichlet functions of arithmetical objects, Thesis Doctor phys.-matem. nauk, Moscow, 1977 (in Russian).Google Scholar
[20] Walsh, J.L., Interpolation and Approximation by Rational Functions in the Complex Domain, Amer. Math. Soc. Coll. Publ., 20 (1960).Google Scholar