De Bruijn-Newman constant: Difference between revisions
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 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 11:50, 25 January 2018
For each real number [math]\displaystyle{ t }[/math], define the entire function [math]\displaystyle{ H_t: {\mathbf C} \to {\mathbf C} }[/math] by the formula
- [math]\displaystyle{ \displaystyle H_t(z) := \int_0^\infty e^{tu^2} \Phi(u) \cos(zu)\ du }[/math]
where [math]\displaystyle{ \Phi }[/math] is the super-exponentially decaying function
- [math]\displaystyle{ \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]\displaystyle{ \Phi }[/math] is even, and that [math]\displaystyle{ H_t }[/math] is even, real on the real axis, and obeys the functional equation [math]\displaystyle{ H_t(\overline{z}) = \overline{H_t(z)} }[/math]. In particular, the zeroes of [math]\displaystyle{ 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]\displaystyle{ \Lambda }[/math], such that [math]\displaystyle{ H_t }[/math] has all zeroes real precisely when [math]\displaystyle{ t \geq \Lambda }[/math]. The Riemann hypothesis is equivalent to the claim that [math]\displaystyle{ \Lambda \leq 0 }[/math]. Currently it is known that [math]\displaystyle{ 0 \leq \Lambda \lt 1/2 }[/math].
