Abstract
The Riemann zeta-function ζ(s) has the property that, for each function g(z) meromorphic on ℂ, there exists a sequence \(s=q_{n}(z)=a_n+b_{n}z\) of linear transformations and an increasing sequence of compact sets K m, whose union is the complex plane ℂ, such that the sequence ζ ◦ q n converges spherically uniformly to g on each K m.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. Andersson, Lavrent’ev’s approximation theorem with nonvanishing polynomials and universality of zeta-functions, in: J. Steuding and R. Steuding (eds.), New Directions in Value-Distribution Theory of Zeta and L-Functions, Proceedings of the conference, Würzburg, Germany, October 6–10, 2008, Shaker Verlag, Aachen, (2009), 7–10.
—, Mergelyan’s approximation theorem with nonvanishing polynomials and universality of zeta-functions, arXiv:1010.0850v1 [math.CV], (5 Oct 2010), 13 pp.
B. Bagchi, A joint universality theorem for Dirichlet L-functions, Math. Z. 181 (1982), 319–334.
R. Fournier and V. Nestoridis, Non-normal sequences of holomorphic functions and universality, Comput. Methods Funct. Theory 11 no.1 (2011), 309–316.
P. M. Gauthier, Approximation of and by the Riemann zeta-function, Comput. Methods Funct. Theory 10 no.2 (2010), 603–638.
P. M. Gauthier, Approximating functions having zeros or poles by the Riemann zeta-function, Comput. Methods Funct. Theory 12 no.1 (2011), 257–271.
J. Steuding, Value Distribution of L-Functions, Lecture Notes in Mathematics 1877, Springer, Berlin, 2007.
Author information
Authors and Affiliations
Corresponding author
Additional information
Partially supported by NSERC (Canada).
Rights and permissions
About this article
Cite this article
Gauthier, P.M. Approximating all Meromorphic Functions by Linear Motions of the Riemann Zeta-Function. Comput. Methods Funct. Theory 12, 517–526 (2012). https://doi.org/10.1007/BF03321841
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF03321841