On the Topology of the Sets of the Real Projections of the Zeros of Exponential Polynomials

  • Gaspar MoraEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 80)


It is analysed some topological properties of the set \(P_{D(z)}\) of the real projections of the zeros of an exponential polynomial \(D(z)\) of the form \(1+\sum _{j=1}^{n}m_{j}\,\mathrm{{e}}^{\omega _{j}z}\), where \(n\) is a positive integer, \(m_{j}\in \mathbb {C}\setminus \left\{ 0\right\} \) and \(\omega _{j}>0\), for all \(1\le j\le n\). It is pointed out the influence of the \(\omega \)’s, called frequencies, as well as the \(m\)’s, called coefficients of \(D(z)\), on the topology of \(P_{D(z)}\). Reciprocally, it is seen how the order of the zeros of \(D(z)\) is also influenced by the topology of \(P_{D(z)}\). Finally, has been proved that the set of the real projections of the zeros of every partial sum \(\zeta _{n}(z):=\sum _{k=1}^{n}1/k^{z}\), \(n>2\), of the Riemann zeta function situated in the critical strip of the Riemann zeta function is dense in itself. Therefore the closure of this set is a perfect set, for all \(n>2\).


Exponential polynomials Zeros of the partial sums of the Riemann zeta function Perfect sets 


  1. 1.
    Apostol, T.M.: Modular Functions and Dirichlet Series in Number Theory. Springer, New York (1990)CrossRefzbMATHGoogle Scholar
  2. 2.
    Avellar, C.E., Hale, J.K.: On the zeros of exponential polynomials. J. Math. Anal. Appl. 73, 434–452 (1980)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Balazard, M., Velásquez-Castañón, O.: Sur l’infimum des parties réelles des zéros des sommes partielles de la fonction zêta de Riemann. C. R. Acad. Sci. Paris Ser. I 347 343–346 (2009)Google Scholar
  4. 4.
    Bohr, H.: Zur theorie der allgemeinen dirichletschen reihen. Math. Ann. 79, 136–156 (1919)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Bohr, H.: Almost Periodic Functions. Chelsea Publishing Company, New York (1947)Google Scholar
  6. 6.
    Borwein, P., Choi, S., Rooney, B., Weirathmueller, A.: The Riemann Hypothesis. Springer, Canada (2008)Google Scholar
  7. 7.
    Dubon, E., Mora, G., Sepulcre, J.M., Ubeda, J.I., Vidal, T.: A note on the real projection of the zeros of partial sums of the Riemann zeta function. RACSAM. doi:  10.1007/s13398-012-0094-2
  8. 8.
    Edwards, H.M.: Riemann’s Zeta Function. Academic Press, New York (1974)zbMATHGoogle Scholar
  9. 9.
    Farag, H.M.: Dirichlet truncations of the riemann zeta function in the critical strip possess zeros near every vertical line. Inter. J. Number Theory 4(4), 653–662 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers. Clarendon Press, Oxford (1954)zbMATHGoogle Scholar
  11. 11.
    Haselgrove, C.B.: A disproof of a conjecture of Pólya. Mathematika 5, 141–145 (1958)Google Scholar
  12. 12.
    Henry, D.: Linear autonomous neutral functional differential equations. J. Differ. Eqn. 15, 106–128 (1974)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Kuratowski, K.: Topology. Academic Press, New York (1966)Google Scholar
  14. 14.
    Langer, R.E.: On the zeros of exponential sums and integrals. Bull. Amer. Math. Soc. 37, 213–239 (1931)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Lapidus, M.L., Pomerance, C.: The riemann zeta-function and the one-dimensional Weyl-Berry conjecture for fractal drums. Proc. London Math. Soc. 66(3), 41–69 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Lapidus, M.L., Van Frankenhuysen, M.: Complex dimensions of self-similar fractal strings and diophantine approximation. Exp. Math. 12, 41–69 (2003)CrossRefzbMATHGoogle Scholar
  17. 17.
    Lapidus, M.L., van Frankenhuysen, M.: Fractality, self-similarity and complex dimensions. In: Proceeding of Symposia in Pure Mathematics 72, Part 1. American Mathematical Society, Providence (2004)Google Scholar
  18. 18.
    Lapidus, M.L., Van Frankenhuysen, M.: Fractal geometry, complex dimensions and zeta functions: geometry and spectra of fractal strings. Springer, New York (2006)Google Scholar
  19. 19.
    Levinson, N.: Asymptotic formula for the coordinates of the zeros of sections of the zeta function, \(\zeta _{N}(s)\), near \(s=1\). Proc. Nat. Acad. Sci. USA 70, 985–987 (1973)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Montgomery, H.L.: Zeros of approximations to the zeta function. In: Studies in Pure Mathematics: To the Memory of Paul Turán, pp. 497–506. Birkhäuser, Basel (1983)Google Scholar
  21. 21.
    Montgomery, H.L., Vaughan, R.C.: Mean values of multiplicative functions. Period. Math. Hungar. 43, 199–214 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Mora, G.: A note on the functional equation \(F(z)+F(2z)+\cdots +F(nz)=0 \). J. Math. Anal. Appl. 340, 466–475 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Mora, G., Sepulcre, J.M.: On the distribution of zeros of a sequence of entire functions approaching the riemann zeta function. J. Math. Anal. Appl. 350, 409–415 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Mora, G., Sepulcre, J.M.: The critical strips of the sums \(1+2^{z}+\cdots +n^{z}\). Abstr. Appl. Anal. 909674, 15 (2011). doi: 10.1155/2011/909674
  25. 25.
    Mora, G., Sepulcre, J.M.: The zeros of riemann zeta partial sums yield solutions to \(f(x)+f(2x)+\cdots +f(nx)=0\). Mediterr. J. Math. 10, 1221–1233 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Mora, G., Sepulcre, J.M.: Privileged regions in critical strips of non-lattice dirichlet polynomials. Complex. Anal. Oper. Theory 7, 1417–1426 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Mora, G.: On the asymptotically uniform distribution of the zeros of the partial sums of the riemann zeta function. J. Math. Anal. Appl. 403, 120–128 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Mora, G., Sepulcre, J.M., Vidal, T.: On the existence of exponential polynomials with prefixed gaps. Bull. London Math. Soc. (2013). doi: 10.1112/blms/bdt043
  29. 29.
    Moreno, J.C.: The zeros of exponential polynomials (I). Compos. Math. 26(1), 69–78 (1973)zbMATHGoogle Scholar
  30. 30.
    Pólya, G., Szëgo, G.: Problems and theorems in analysis, vol. II. Springer, New York (1976)Google Scholar
  31. 31.
    Ritt, J.F.: On the zeros of exponential polynomials. Trans. Amer. Math. Soc. 31, 680–686 (1929)CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Spira, R.: Zeros of sections of the zeta function. I. Math. Comp. 20, 542–550 (1966)CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Spira, R.: Zeros of sections of the zeta function. II. Math. Comp. 22, 168–173 (1968)MathSciNetGoogle Scholar
  34. 34.
    Tamarkin, J.D.: The zeros of certain integral functions. J. London Math. Soc. 2, 66–69 (1927)CrossRefzbMATHMathSciNetGoogle Scholar
  35. 35.
    Tamarkin, J.D.: Some general problems of the theory of ordinary linear differential equations and expansions of an arbitrary function in series of fundamental functions. Math. Z. 27, 1–54 (1928)CrossRefMathSciNetGoogle Scholar
  36. 36.
    Titchmarsh, E.C.: The Theory of Functions, 2nd edn. Oxford University Press, London (1939)zbMATHGoogle Scholar
  37. 37.
    Turán, P.: On some approximate dirichlet polynomials in the theory of the zeta-function of riemann. Dansk. Vid. Selsk. Mat.-Fys Medd. 24(17), 3–36 (1948)Google Scholar
  38. 38.
    Voronin, S.M.: On the zeros of partial sums of the dirichlet series for the riemann zeta-function. Dolk. Akad. Nauk. SSSR. 216, 964–967 (1974) (Trans. Soviet. Math. Doklady 15, 900–903, 1974)Google Scholar
  39. 39.
    Wilder, C.E.: Expansion problems of ordinary linear differential equations with auxiliary conditions at more than two points. Trans. Amer. Math. Soc. 18, 415–442 (1917)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Department of Mathematical AnalysisUniversity of AlicanteAlicanteSpain

Personalised recommendations