An improved bound for the de Bruijn-Newman constant

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Abstract

The Riemann hypothesis is equivalent to the conjecture that the de Bruijn-Newman constant Λ satisfies Λ ≤ 0. However, so far all the bounds that have been proved for Λ go in the other direction, and provide support for the conjecture of Newman that Λ ≥ 0. This paper shows how to improve previous lower bounds and prove that -2.7 · 10-9 < Λ. This can be done using a pair of zeros of the Riemann zeta function near zero number 1020 that are unusually close together. The new bound provides yet more evidence that the Riemann hypothesis, if true, is just barely true.

Original languageEnglish (US)
Pages (from-to)293-303
Number of pages11
JournalNumerical Algorithms
Volume25
Issue number1-4
DOIs
StatePublished - 2000

Keywords

  • De Bruijn-Newman constant
  • Lehmer pair
  • Riemann hypothesis
  • Zeta function

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