On the de Bruijn-Newman constant

Haseo Ki, Young One Kim, Jungseob Lee

Research output: Contribution to journalArticle

7 Citations (Scopus)

Abstract

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
Volume222
Issue number1
DOIs
Publication statusPublished - 2009 Sep 10

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Riemann hypothesis
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Entire Function
Riemann zeta function
Strip
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All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Cite this

Ki, Haseo ; Kim, Young One ; Lee, Jungseob. / On the de Bruijn-Newman constant. In: Advances in Mathematics. 2009 ; Vol. 222, No. 1. pp. 281-306.
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On the de Bruijn-Newman constant. / Ki, Haseo; Kim, Young One; Lee, Jungseob.

In: Advances in Mathematics, Vol. 222, No. 1, 10.09.2009, p. 281-306.

Research output: Contribution to journalArticle

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