## 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 |
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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 |

### 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)