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

Abstract

We give here a rigorous formulation for a pair of consecutive simple positive zeros of the functionH 0 (which is closely related to the Riemann ξ-function) to be a “Lehmer pair” of zeros ofH 0. 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:

$$ - 4.379 \cdot 10^{ - 6}< \Lambda $$

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Dedicated to the memory of Professor Derrick H. Lehmer

Communicated by Dietrich Braess.

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Csordas, G., Smith, W. & Varga, R.S. Lehmer pairs of zeros, the de Bruijn-Newman constant Λ, and the Riemann Hypothesis. Constr. Approx 10, 107–129 (1994). https://doi.org/10.1007/BF01205170

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AMS classification

  • 30D10
  • 30D15

Key words and phrases

  • Riemann Hypothesis
  • de Bruijn-Newman constant
  • Lehmer pairs of zeros