The Ramanujan Journal

, Volume 46, Issue 1, pp 245–267 | Cite as

Almost periodic functions in terms of Bohr’s equivalence relation



In this paper we introduce an equivalence relation on the classes of almost periodic functions of a real or complex variable which is used to refine Bochner’s result that characterizes these spaces of functions. In fact, with respect to the topology of uniform convergence, we prove that the limit points of the family of translates of an almost periodic function are precisely the functions which are equivalent to it, which leads us to a characterization of almost periodicity. In particular we show that any exponential sum which is equivalent to the Riemann zeta function, \(\zeta (s)\), can be uniformly approximated in \(\{s=\sigma +it:\sigma >1\}\) by certain vertical translates of \(\zeta (s)\).


Almost periodic functions Exponential sums Riemann zeta function Bochner’s theorem Fourier series Dirichlet series 

Mathematics Subject Classification

30D20 30B50 11K60 30Axx 



The authors thank the anonymous referee for his/her valuable comments on our manuscript which led us to generalize our results.


  1. 1.
    Apostol, T.M.: Modular Functions and Dirichlet Series in Number Theory. Springer, New York (1990)CrossRefMATHGoogle Scholar
  2. 2.
    Besicovitch, A.S.: Almost Periodic Functions. Dover, New York (1954)MATHGoogle Scholar
  3. 3.
    Bochner, S.: A new approach to almost periodicity. Proc. Natl. Acad. Sci. 48, 2039–2043 (1962)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bohr, H.: Zur Theorie der fastperiodischen Funktionen. (German) III. Dirichletentwicklung analytischer Funktionen. Acta Math. 47(3), 237–281 (1926)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bohr, H.: Almost Periodic Functions. Chelsea, New York (1951)MATHGoogle Scholar
  6. 6.
    Bohr, H.: Contribution to the theory of almost periodic functions, Det Kgl. danske Videnskabernes Selskab. Matematisk-fisiske meddelelser. Bd. XX. Nr. 18, Copenhague (1943)Google Scholar
  7. 7.
    Corduneanu, C.: Almost Periodic Functions. Interscience Publishers, New York (1968)MATHGoogle Scholar
  8. 8.
    Corduneanu, C.: Almost Periodic Oscillations and Waves. Springer, New York (2009)CrossRefMATHGoogle Scholar
  9. 9.
    Favorov, S.Y.U.: Zeros of holomorphic almost periodic functions. J. Anal. Math. 84, 51–66 (2001)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Fink, A.M.: Almost Periodic Differential Equations. Lecture Notes in Mathematics, vol. 377. Springer, New York (1974)MATHGoogle Scholar
  11. 11.
    Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers. Oxford University Press, Oxford (1979)MATHGoogle Scholar
  12. 12.
    Jessen, B.: Some aspects of the theory of almost periodic functions. In: Proceedings of International Congress Mathematicians Amsterdam, 1954, Vol. 1. North-Holland, pp. 304–351 (1954)Google Scholar
  13. 13.
    Karatsuba, A.A., Voronin, S.M.: The Riemann Zeta Function. Walter de Gruyter & Co., Berlin (1992)CrossRefMATHGoogle Scholar
  14. 14.
    Laurinčikas, A.: Universality of the Riemann zeta-function. J. Number Theory 130, 2323–2331 (2010)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Laurinčikas, A., Schwarz, W., Steuding, J.: The universality of general Dirichlet series. Analysis (Munich) 23(1), 13–26 (2003)MathSciNetMATHGoogle Scholar
  16. 16.
    Lehman, R.S.: On Liouville’s function. Math. Comput. 14, 311–320 (1960)MathSciNetMATHGoogle Scholar
  17. 17.
    Sepulcre, J.M., Vidal, T.: Equivalence classes of exponential polynomials with the same set of zeros. Complex Var. Elliptic Equ. 61(2), 225–238 (2016)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Titchmarsh, E.C.: The Theory of the Riemann Zeta-Function, 2nd edn. Oxford Science Publication, London (1986)MATHGoogle Scholar
  19. 19.
    Titchmarsh, E.C.: The Theory of Functions, 2nd edn. Oxford University Press, London (1976)MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of AlicanteAlicanteSpain

Personalised recommendations