SpringerLink
Log in

Multiple Positivity and the Riemann Zeta-Function

  • Published:
Computational Methods and Function Theory Aims and scope Submit manuscript

Abstract

This paper investigates the relationship between the Riemann ζ-function and various classes of generating functions of multiply positive sequences introduced by Schoenberg.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (Germany)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. Aissen, A. Edrei, I. J. Schoenberg and A. Whitney, On the generating functions of totally positive sequences, J. Analyse Math. 2 (1953), 93–109.

    Article  Google Scholar 

  2. T. Craven and G. Csordas, Jensen polynomials and the Turan and Laguerre inequalities, Pac. J. Math. 136 (1989), 241–260.

    Article  MathSciNet  MATH  Google Scholar 

  3. —, Iterated Laguerre and Turan inequalities, J. Inequal. Pure Appl. Math. 3 (2002) no.3, article 39, 14 pp. (electronic).

  4. G. Csordas, T. S. Norfolk and R. S. Varga, The Riemann hypothesis and the Turan inequalities, Trans. Amer. Math. Soc. 296 (1986), 521–541.

    MathSciNet  MATH  Google Scholar 

  5. G. Csordas and R. S. Varga, Moment inequalities and the Riemann Hypothesis, Constr. Approx. 4 (1988), 175–198.

    Article  MathSciNet  MATH  Google Scholar 

  6. G. Csordas and R. S. Varga, Necessary and sufficient conditions and the Riemann Hypothesis, Adv. In Appl. Math. 1 (1990), 328–357.

    Article  MathSciNet  Google Scholar 

  7. G. Csordas, Complex zero decreasing sequences and the Riemann Hypothesis II, in: H. G. W. Begehr et al. (ed.), Analysis and Applications-ISA AC 2001, Proceedings of the 3rd international congress, Berlin, Germany, August 20–25, 2001, Dordrecht, Kluwer Academic Publishers. Int. Soc. Anal. Appl. Comput. 10, (2003), 121–134.

    Google Scholar 

  8. G. Csordas and C.-C. Yang, On the zeros of the Riemann ξ-function, Southwest J. Pure Appl. Math. 1 (2003), 33–42.

    MathSciNet  Google Scholar 

  9. D. K. Dimitrov, Higher order Turan ineqalities, Proc. Amer. Math. Soc. 126 (1998), 2033–2037.

    Article  MathSciNet  MATH  Google Scholar 

  10. H. M. Edwards, Riemanns Zeta Function, Academic Press, New York — London 1974.

    Google Scholar 

  11. M. Fekete and G. Pólya, Über ein Problem von Laguerre, Rend. Circ. Mat. Palermo 34 (1912), 89–120.

    Article  MATH  Google Scholar 

  12. A. A. Gol’dberg and I. V. Ostrovskii, Indicators of entire Hermitian-positive functions of finite order, (in Russian), Sib. Math. Zh. 23 1982) No.6, 55–73; English transl. in: Sib. Math. J. 23 (1983), 804–820.

    MathSciNet  Google Scholar 

  13. A. A. Gol’dberg and I. V. Ostrovskii, Indicators of entire absolutely monotonic functions of finite order, (in Russian), Sib. Math. Zh. 27 1986) No.6, 33–49.

    MathSciNet  MATH  Google Scholar 

  14. A. E. Ingham, The Distribution of Prime Numbers, Cambridge University Press, Cambridge 1932.

    Google Scholar 

  15. A. A. Karatsuba, Osnovy analiticheskoj teorii chisel, 3-e izd. (in Russian, Principles of Analytic Number Theory, 3-rd ed.), Moskva: URSS 2004.

  16. S. Karlin, Total Positivity Vol. I, Stanford University Press, California 1968.

    MATH  Google Scholar 

  17. O. M. Katkova and I. V. Ostrovskii, Zero sets of entire generating functions of Pólya frequency sequences of finite order, (in Russian), Izv. Akad. Nauk SSSR, Ser. Mat 53 no.4, (1989), 771–781; Englisch transl. in: Math. USSR, Izv. 35 (1990), 101–112.

    MATH  Google Scholar 

  18. O. M. Katkova, On the growth of entire generating functions of multiply positive sequences, Mat. Fiz. Anal. Geom. 10 (2003), 61–75.

    MathSciNet  MATH  Google Scholar 

  19. O. M. Katkova, Entire functions with asymptotically multiply positive sequence of coefficients, (in Russian), Teoriya Funktsii, Funktsional. Anal. i Prolozhen. 49 (1988), 51–59; English transl.: Journal of Soviet Math. 49 no.4 (1990), 1070–1075.

    MathSciNet  MATH  Google Scholar 

  20. O. M. Katkova, On a certain method of construction of the Pólya frequency sequences, (in Russian), Teoriya Funktsii, Funktsional. Anal. i Prilozhen. 51 (1989), 129–137, English transl.: Journal of Soviet Math. bd]52 no.6 (1990), 3539–3545.

    MATH  Google Scholar 

  21. E. Laguerre, Oeuvres Vol. I, Paris, Gauthier-Villars 1898.

    Google Scholar 

  22. B. Ya. Levin, Raspredelenye nuley tselykh funktsiy, (in Russian), Gosudarstvennoye Iz-datiel’stvo Tekhniko-Teoriticheskoy Literatury, Moskva 1956; English transl.: Distribution of Zeros of Entire Functions, Transl. Math. Monographs, vol.5, AMS, Providence, RI 1980.

  23. A. M. Odlyzko, The 1022-th zero of the Riemann zeta function, in: M. van Frankenhuysen and M. L. Lapidus (eds.), Dynamical, Spectral, and Arithmetic Zeta Functions 290, Contemporary Math., Amer. Math. Soc, 2001, 139–144.

  24. G. Pólya, Über einen Satz von Laguerre, Jber. Deutsch. Math.-Verein. 38 (1929), 161–168.

    MATH  Google Scholar 

  25. G. Pólya, Über die algebraisch-funktionentheoretischen Untersuchungen von J. L. W. V. Jensen, Kgl. Danske Vid. Sel. Math.-Fys. Medd. 7 (1927), 3–33.

    Google Scholar 

  26. G. Pólya, Collected Papers Vol. II: Location of Zeros, (R. P. Boas ed.), MIT Press, Cambridge, MA 1974.

  27. G. Pólya and J. Schur, Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen, J. Reine Angew. Math. 144 (1914), 89–113.

    MATH  Google Scholar 

  28. G. Pólya and G. Szegö, Aufgaben und Lehrsätze aus der Analysis II, Springer Verlag, Berlin 1964.

    Google Scholar 

  29. G. A. Postnikov, Vvedenie v analititcheskuju teoriju tchisel, Nauka, Moskva 1971; English transl.: Introduction to Analytic Number Theory, Translation of Mathematical Monographs 68, RI: American Mathematical Society (AMS), VI, Providence 1988.

    Google Scholar 

  30. I. J. Schoenberg, On the zeros of the generating functions of multiply positive sequences and functions, Ann. of Math. 62 (1955), 447–471.

    Article  MathSciNet  MATH  Google Scholar 

  31. I. J. Schoenberg, A note on multiply positive sequences and the Descartes rule of signs, Rend. Corc. Mat. Palermo (2) 4 (1955), 123–131.

    Article  MathSciNet  MATH  Google Scholar 

  32. E. S. Titchmarsh, The Theory of the Riemann zeta-Function, Clarendon Press, Oxford, 1951.

    MATH  Google Scholar 

  33. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, University Press, Cambridge, 1927.

    MATH  Google Scholar 

  34. X. Gourdon, The 1013 first zeros of the Riemann zeta function, and zeros computation at very large height, http://numbers.computation.free.fr/Constants/Miscellaneous/zetazeros1e13-1e24.pdf, October 2004.

Download references

Author information

Authors and Affiliations

  1. Dept. of Math., Kharkov National University, Svobody sq., 4, 61077, Kharkov, Ukraine

    Olga M. Katkova

Authors
  1. Olga M. Katkova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Olga M. Katkova.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Katkova, O.M. Multiple Positivity and the Riemann Zeta-Function. Comput. Methods Funct. Theory 7, 13–31 (2007). https://doi.org/10.1007/BF03321628

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03321628

Keywords

2000 MSC

Search

Navigation