Difference between revisions of "De Bruijn-Newman constant"

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(Created page with "For each real number <math>t</math>, define the entire function <math>H_t: {\mathbf C} \to {\mathbf C}</math> by the formula :<math>\displaystyle H_t(z) := \int_0^\infty e^{t...")
 
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It is known that <math>\Phi</math> is even, and that <math>H_t</math> is even, real on the real axis, and obeys the functional equation <math>H_t(\overline{z}) = \overline{H_t(z)}</math>.  In particular, the zeroes of <math>H_t</math> are symmetric about both the real and imaginary axes.
 
It is known that <math>\Phi</math> is even, and that <math>H_t</math> is even, real on the real axis, and obeys the functional equation <math>H_t(\overline{z}) = \overline{H_t(z)}</math>.  In particular, the zeroes of <math>H_t</math> are symmetric about both the real and imaginary axes.
  
De Bruijn and Newman showed that there existed a constant, the _de Bruijn-Newman constant_ <math>\Lambda</math>, such that <math>H_t</math> has all zeroes real precisely when <math>t \geq \Lambda</math>.  The Riemann hypothesis is equivalent to the claim that <math>\Lambda \leq 0</math>.  Currently it is known that <math>0 \leq \Lambda < 1/2</math>.
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De Bruijn and Newman showed that there existed a constant, the ''de Bruijn-Newman constant'' <math>\Lambda</math>, such that <math>H_t</math> has all zeroes real precisely when <math>t \geq \Lambda</math>.  The Riemann hypothesis is equivalent to the claim that <math>\Lambda \leq 0</math>.  Currently it is known that <math>0 \leq \Lambda < 1/2</math>.

Revision as of 12:50, 25 January 2018

For each real number [math]t[/math], define the entire function [math]H_t: {\mathbf C} \to {\mathbf C}[/math] by the formula

[math]\displaystyle H_t(z) := \int_0^\infty e^{tu^2} \Phi(u) \cos(zu)\ du[/math]

where [math]\Phi[/math] is the super-exponentially decaying function

[math]\displaystyle \Phi(u) := \sum_{n=1}^\infty (2\pi^2 n^4 e^{9u} - 3 \pi n^2 e^{5u}) \exp(-\pi n^2 e^{4u}).[/math]

It is known that [math]\Phi[/math] is even, and that [math]H_t[/math] is even, real on the real axis, and obeys the functional equation [math]H_t(\overline{z}) = \overline{H_t(z)}[/math]. In particular, the zeroes of [math]H_t[/math] are symmetric about both the real and imaginary axes.

De Bruijn and Newman showed that there existed a constant, the de Bruijn-Newman constant [math]\Lambda[/math], such that [math]H_t[/math] has all zeroes real precisely when [math]t \geq \Lambda[/math]. The Riemann hypothesis is equivalent to the claim that [math]\Lambda \leq 0[/math]. Currently it is known that [math]0 \leq \Lambda \lt 1/2[/math].