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On the Mishou Theorem for Zeta-Functions with Periodic Coefficients

Author

Listed:
  • Aidas Balčiūnas

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

  • Mindaugas Jasas

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

  • Renata Macaitienė

    (Institute of Regional Development, Šiauliai Academy, Vilnius University, P. Višinskio Str. 25, LT-76351 Šiauliai, Lithuania
    These authors contributed equally to this work.)

  • Darius Šiaučiūnas

    (Institute of Regional Development, Šiauliai Academy, Vilnius University, P. Višinskio Str. 25, LT-76351 Šiauliai, Lithuania
    These authors contributed equally to this work.)

Abstract

Let a = { a m } and b = { b m } be two periodic sequences of complex numbers, and, additionally, a is multiplicative. In this paper, the joint approximation of a pair of analytic functions by shifts ( ζ n T ( s + i τ ; a ) , ζ n T ( s + i τ , α ; b ) ) of absolutely convergent Dirichlet series ζ n T ( s ; a ) and ζ n T ( s , α ; b ) involving the sequences a and b is considered. Here, n T → ∞ and n T ≪ T 2 as T → ∞ . The coefficients of these series tend to a m and b m , respectively. It is proved that the set of the above shifts in the interval [ 0 , T ] has a positive density. This generalizes and extends the Mishou joint universality theorem for the Riemann and Hurwitz zeta-functions.

Suggested Citation

  • Aidas Balčiūnas & Mindaugas Jasas & Renata Macaitienė & Darius Šiaučiūnas, 2023. "On the Mishou Theorem for Zeta-Functions with Periodic Coefficients," Mathematics, MDPI, vol. 11(9), pages 1-10, April.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:9:p:2042-:d:1132692
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