The Laguerre inequalities with applications to a problem associated with the Riemann hypothesis

1572-9265

AMS (MOS): 30D10 ; 30D15 ; 65E05 ; CR:G1.m ; Laguerre inequalities ; Riemann Hypothesis ; de Bruijn-Newman constant

Springer Online Journal Archives 1860-2000

Computer Science

Mathematics

Abstract We investigate here a new numerical method, base on the Laguerre inequalities, for determining lower bounds for the de Bruijn-Newman constant ∧, which is related to the Riemann Hypothesis. (Specifically, the truth of the Riemann Hypothesis would imply that ∧≦0.) Unlike previous methods which involved either finding nonreal zeros of associated Jensen polynomials or finding nonreal zeros of a certain real entire function, this new method depends only on evaluating, in real arithmetic, the Laguerre difference $$L_1 (H_\lambda (x))\begin{array}{*{20}c} {\text{.}} \\ {\text{.}} \\ \end{array} = (H'_\lambda (x))^2 - H_\lambda (x) \cdot H''_{_\lambda } (x){\text{ (}}x,{\text{ }}\lambda \in \mathbb{R}{\text{)}}$$ where $$(H_\lambda (z)\begin{array}{*{20}c} {\text{.}} \\ {\text{.}} \\ \end{array} = \int_0^\infty {e^{\lambda t^2 } \Phi (t)}$$ cos(tz)dt is a real entire function. We apply this method to obtain the new lower bound for ∧, -0.0991 〈 ∧ which improves all previously published lower bounds for ∧.

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