# Difference between revisions of "De Bruijn-Newman constant"

For each real number $t$, define the entire function $H_t: {\mathbf C} \to {\mathbf C}$ by the formula
$\displaystyle H_t(z) := \int_0^\infty e^{tu^2} \Phi(u) \cos(zu)\ du$
where $\Phi$ is the super-exponentially decaying function
$\displaystyle \Phi(u) := \sum_{n=1}^\infty (2\pi^2 n^4 e^{9u} - 3 \pi n^2 e^{5u}) \exp(-\pi n^2 e^{4u}).$
It is known that $\Phi$ is even, and that $H_t$ is even, real on the real axis, and obeys the functional equation $H_t(\overline{z}) = \overline{H_t(z)}$. In particular, the zeroes of $H_t$ are symmetric about both the real and imaginary axes.
De Bruijn and Newman showed that there existed a constant, the de Bruijn-Newman constant $\Lambda$, such that $H_t$ has all zeroes real precisely when $t \geq \Lambda$. The Riemann hypothesis is equivalent to the claim that $\Lambda \leq 0$. Currently it is known that $0 \leq \Lambda \lt 1/2$.