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Constructive Approximation

, Volume 10, Issue 1, pp 107–129 | Cite as

Lehmer pairs of zeros, the de Bruijn-Newman constant Λ, and the Riemann Hypothesis

  • George Csordas
  • Wayne Smith
  • Richard S. Varga
Article

Abstract

We give here a rigorous formulation for a pair of consecutive simple positive zeros of the functionH0 (which is closely related to the Riemann ξ-function) to be a “Lehmer pair” of zeros ofH0. With this formulation, we establish that each such pair of zeros gives a lower bound for the de Bruijn-Newman constant Λ (where the Riemann Hypothesis is equivalent to the assertion that Λ≤0). We also numerically obtain the following new lower bound for Λ:
$$ - 4.379 \cdot 10^{ - 6}< \Lambda $$

AMS classification

30D10 30D15 

Key words and phrases

Riemann Hypothesis de Bruijn-Newman constant Lehmer pairs of zeros 

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References

  1. [B]N. G. de Bruijn (1950):The roots of trigonometric integrals. Duke J. Math.,17:197–226.Google Scholar
  2. [CNV1]G. Csordas, T. S. Norfolk, R. S. Varga (1986).The Riemann Hypothesis and the Turán inequalities. Trans. Amer. Math. Soc.,296:521–541.Google Scholar
  3. [CNV2]G. Csordas, T. S. Norfolk, R. S. Varga (1988):A lower bound for the de Bruijn-Newman constant Λ. Numer. Math.,52:483–497.Google Scholar
  4. [CRV]G. Csordas, A. Ruttan, R. S. Varga (1991):The Laguerre inequalities with applications to a problem associated with the Riemann Hypothesis. Numer. Algorithms,1:305–330.Google Scholar
  5. [CV]G. Csordas, R. S. Varga (1990):Necessary and sufficient conditions and the Riemann Hypothesis. Adv. in Appl. Math.,11:328–357.Google Scholar
  6. [E]H. M. Edwards (1974): Riemann's Zeta Function. New York: Academic Press.Google Scholar
  7. [L]D. H. Lehmer (1956):On the roots of the Riemann zeta-function. Acta Math.,95:291–298.Google Scholar
  8. [LRW]J. van de Lune, H. J. J. te Riele, D. T. Winter (1986):On the zeros of the Riemann zeta function in the critical strip, IV. Math. Comp.46:667–681.Google Scholar
  9. [N]C. M. Newman (1976):Fourier transforms with only real zeros. Proc. Amer. Math. Soc.,61:245–251.Google Scholar
  10. [NRV]T. S. Norfolk, A. Ruttan, R. S. Varga (1992):A lower bound for the de Bruijn-Newman constant Λ, II. In: Progress in Approximation Theory (A. A. Gonchar, E. B. Saff, eds.). New York: Springer-Verlag, pp. 403–418.Google Scholar
  11. [Ob]N. Obreschkoff (1963): Verteilung und Berechnung der Nullstellen reeler Polynome. Berlin: VEB.Google Scholar
  12. [Od]A. M. Odlyzko: Personal communication.Google Scholar
  13. [R1]H. J. J. te Riele (1979): Table of the First 15,000 Zeros of the Riemann Zeta Function to 28 Significant Digits and Related Quantities. Report Number 67/79, Mathematisch Centrum, Amsterdam.Google Scholar
  14. [R2]H. J. J. te Riele (1991):A new lower bound for the de Bruijn-Newman constant. Numer. Math.,58:661–667.Google Scholar
  15. [T]E. C. Titchmarsh (1986): The Theory of the Riemann Zeta-Function, 2nd edn. (revised by D. R. Heath-Brown). Oxford: Oxford University Press.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1994

Authors and Affiliations

  • George Csordas
    • 1
  • Wayne Smith
    • 1
  • Richard S. Varga
    • 2
  1. 1.Department of MathematicsUniversity of HawaiiHonoluluUSA
  2. 2.Institute for Computational MathematicsKent State UniversityKentUSA

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