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


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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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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