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On Joint Universality in the Selberg–Steuding Class

Author

Listed:
  • Roma Kačinskaitė

    (Institute of Mathematics, Faculty of Mathematics and Informatics, Vilnius University, Naugarduko Str. 24, LT-03225 Vilnius, Lithuania
    These authors contributed equally to this work.)

  • Antanas Laurinčikas

    (Institute of Mathematics, Faculty of Mathematics and Informatics, Vilnius University, Naugarduko Str. 24, LT-03225 Vilnius, Lithuania
    These authors contributed equally to this work.)

  • Brigita Žemaitienė

    (Institute of Mathematics, Faculty of Mathematics and Informatics, Vilnius University, Naugarduko Str. 24, LT-03225 Vilnius, Lithuania
    These authors contributed equally to this work.)

Abstract

The famous Selberg class is defined axiomatically and consists of Dirichlet series satisfying four axioms (Ramanujan hypothesis, analytic continuation, functional equation, multiplicativity). The Selberg–Steuding class S is a complemented Selberg class by an arithmetic hypothesis related to the distribution of prime numbers. In this paper, a joint universality theorem for the functions L from the class S on the approximation of a collection of analytic functions by shifts L ( s + i a 1 τ ) , … , L ( s + i a r τ ) , where a 1 , … , a r are real algebraic numbers linearly independent over the field of rational numbers, is obtained. It is proved that the set of the above approximating shifts is infinite, its lower density and, with some exception, density are positive. For the proof, a probabilistic method based on weak convergence of probability measures in the space of analytic functions is applied together with the Backer theorem on linear forms of logarithms and the Mergelyan theorem on approximation of analytic functions by polynomials.

Suggested Citation

  • Roma Kačinskaitė & Antanas Laurinčikas & Brigita Žemaitienė, 2023. "On Joint Universality in the Selberg–Steuding Class," Mathematics, MDPI, vol. 11(3), pages 1-15, February.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:3:p:737-:d:1053796
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    References listed on IDEAS

    as
    1. Roma Kačinskaitė & Kohji Matsumoto, 2015. "The mixed joint universality for a class of zeta-functions," Mathematische Nachrichten, Wiley Blackwell, vol. 288(16), pages 1900-1909, November.
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