On the de Bruijn-Newman constant

Haseo Ki, Young One Kim, Jungseob Lee

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11 Citations (Scopus)


If λ(0) denotes the infimum of the set of real numbers λ such that the entire function Ξλ represented byΞλ (t) = underover(∫, 0, ∞) efrac(λ, 4) (log x)2 + frac(i t, 2) log x (x5 / 4 underover(∑, n = 1, ∞) (2 n4 π2 x - 3 n2 π) e- n2 π x) frac(d x, x) has only real zeros, then the de Bruijn-Newman constant Λ is defined as Λ = 4 λ(0). The Riemann hypothesis is equivalent to the inequality Λ ≤ 0. The fact that the non-trivial zeros of the Riemann zeta-function ζ lie in the strip {s : 0 < Re s < 1} and a theorem of de Bruijn imply that Λ ≤ 1 / 2. In this paper, we prove that all but a finite number of zeros of Ξλ are real and simple for each λ > 0, and consequently that Λ < 1 / 2.

Original languageEnglish
Pages (from-to)281-306
Number of pages26
JournalAdvances in Mathematics
Issue number1
Publication statusPublished - 2009 Sept 10

Bibliographical note

Funding Information:
* Corresponding author. E-mail addresses: haseo@yonsei.ac.kr (H. Ki), kimyo@math.snu.ac.kr (Y.-O. Kim), jslee@ajou.ac.kr (J. Lee). 1 H. Ki was supported by the Korea Science and Engineering Foundation (KOSEF) grant funded by the Korea Government (MOST) (No. R01-2007-000-20018-0).

All Science Journal Classification (ASJC) codes

  • Mathematics(all)


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