I'm not an expert in analytic number theory, but I've been curious in this last period about it. I stumbled upon de Bruijn-Newman constant during my researches on Wikipedia, where it is written that
The constant is closely connected with Riemann's hypothesis concerning the zeros of the Riemann zeta-function: since the Riemann hypothesis is equivalent to the claim that all the zeroes of H(0, z) are real, the Riemann hypothesis is equivalent to the conjecture that $ Λ ≤ 0. $
Where
$$H(\lambda, z):=\int_{0}^{\infty} e^{\lambda u^{2}} \Phi(u) \cos (z u) d u$$ $$\Phi(u) = \sum_{n=1}^{\infty} (2\pi^2n^4e^{9u} - 3 \pi n^2 e^{5u} ) e^{-\pi n^2 e^{4u}}$$
Now, I understand that explaining exactly how the constant is connected to RH might be kinda complicated, but what's exactly $\Lambda$?
Why RH is equivalent to $ Λ ≤ 0 $?
Odlyzko (2000) proved that $-2.7\cdot10^{-9}<\Lambda$ and stated that
The new bound provides yet more evidence that the Riemann Hypothesis, if true, is just barely true.
What does he mean by "barely true"?
Thanks.