Skip to main content

Approximating the Riemann Zeta-Function by Strongly Recurrent Functions

  • Chapter
Blaschke Products and Their Applications

Part of the book series: Fields Institute Communications ((FIC,volume 65))

Abstract

Bhaskar Bagchi has shown that the Riemann hypothesis holds if and only if the Riemann zeta-function ζ(z) is strongly recurrent in the strip 1/2<ℜz<1. In this note we show that ζ(z) can be approximated by strongly recurrent functions sharing important properties with ζ(z).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Bagchi, B.: A joint universality theorem for Dirichlet L-functions. Math. Z. 181, 319–334 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bagchi, B.: Recurrence in topological dynamics and the Riemann hypothesis. Acta Math. Hung. 50, 227–240 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bayart, F., Grivaux, S.: Frequently hypercyclic operators. Trans. Am. Math. Soc. 358(11), 5083–5117 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bayart, F., Matheron, É.: Dynamics of Linear Operators. Cambridge Tracts in Mathematics, vol. 179. Cambridge University Press, Cambridge (2009)

    Book  MATH  Google Scholar 

  5. Chee, P.S.: Universal functions in several complex variables. J. Aust. Math. Soc., Ser. A 28, 189–196 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  6. Gaier, D.: Lectures on Complex Approximation, vol. XV. Birkhäuser, Boston (1987). Transl. from the German by Renate McLaughlin

    Book  MATH  Google Scholar 

  7. Gauthier, P.M., Xiao, J.: The existence of universal inner functions on the unit ball of ℂn. Can. Math. Bull. 48(3), 409–413 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gottschalk, W.H., Hedlund, G.A.: Topological Dynamics. Colloquium Publications of the American Mathematical Society (AMS), vol. 36. American Mathematical Society (AMS), Providence (1955). VIII

    MATH  Google Scholar 

  9. Grosse-Erdmann, K.-G.: Universal families and hypercyclic operators. Bull. Am. Math. Soc., New Ser. 36(3), 345–381 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hamburger, H.: Über einige Beziehungen, die mit der Funktionalgleichung der Riemannschen ζ-Funktion äquivalent sind. Math. Ann. 85, 129–140 (1922)

    Article  MathSciNet  MATH  Google Scholar 

  11. Heins, M.: A universal Blaschke product. Arch. Math. 6, 41–44 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  12. Nieß, M.: Universal approximants of the Riemann zeta-function. Comput. Methods Funct. Theory 9(1), 145–159 (2009)

    MathSciNet  MATH  Google Scholar 

  13. Nieß, M.: Close universal approximants of the Riemann zeta-function. In: Steuding, R., et al. (eds.) Proceedings of the Conference New Directions in Value-Distribution Theory of Zeta and L-Functions, Würzburg, Germany, October 6–10, 2008, pp. 295–303. Shaker Verlag, Aachen (2009)

    Google Scholar 

  14. Nieß, M.: On universal relatives of the Riemann zeta-function. J. Contemp. Math. Anal., Armen. Acad. Sci. 44(5), 335–339 (2009); translation from Izv. Nats. Akad. Nauk Armen., Mat. (5), 83–88 (2009)

    MATH  Google Scholar 

  15. Seidel, W., Walsh, J.L.: On approximation by euclidean and non-euclidean translations of an analytic function. Bull. Am. Math. Soc. 47, 916–920 (1941)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

Supported in part by NSERC (Canada).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to P. M. Gauthier .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer Science+Business Media New York

About this chapter

Cite this chapter

Gauthier, P.M. (2013). Approximating the Riemann Zeta-Function by Strongly Recurrent Functions. In: Mashreghi, J., Fricain, E. (eds) Blaschke Products and Their Applications. Fields Institute Communications, vol 65. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-5341-3_2

Download citation

Publish with us

Policies and ethics