View source for De Bruijn-Newman constant

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 < 1/2</math>.