### Abstract

If λ^{(0)} denotes the infimum of the set of real numbers λ such that the entire function Ξ_{λ} represented byΞ_{λ} (t) = underover(∫, 0, ∞) e^{frac(λ, 4) (log x)2 + frac(i t, 2) log x} (x^{5 / 4} underover(∑, n = 1, ∞) (2 n^{4} π^{2} x - 3 n^{2} 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 Ξ_{λ} are real and simple for each λ > 0, and consequently that Λ < 1 / 2.

Original language | English |
---|---|

Pages (from-to) | 281-306 |

Number of pages | 26 |

Journal | Advances in Mathematics |

Volume | 222 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2009 Sep 10 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Advances in Mathematics*,

*222*(1), 281-306. https://doi.org/10.1016/j.aim.2009.04.003

}

*Advances in Mathematics*, vol. 222, no. 1, pp. 281-306. https://doi.org/10.1016/j.aim.2009.04.003

**On the de Bruijn-Newman constant.** / Ki, Haseo; Kim, Young One; Lee, Jungseob.

Research output: Contribution to journal › Article

TY - JOUR

T1 - On the de Bruijn-Newman constant

AU - Ki, Haseo

AU - Kim, Young One

AU - Lee, Jungseob

PY - 2009/9/10

Y1 - 2009/9/10

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=67349251496&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=67349251496&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2009.04.003

DO - 10.1016/j.aim.2009.04.003

M3 - Article

AN - SCOPUS:67349251496

VL - 222

SP - 281

EP - 306

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

IS - 1

ER -