In a video lecture on Youtube, ''Vaporizing and freezing the Riemann zeta function'', Terry Tao states that ''*de Bruijn proved that if for some $t_0$ the zeros of $H_{t_0}$ are contained in the strip $\lbrace x+iy, |y|\leq y_0\rbrace$, then the zeros of $H_t$ are all real for $t\geq t_0 + \frac{1}{2}y_{0}^2$*'', where $H_{t}(z)=\int_{0}^{\infty} e^{tu^2}\phi(u)\cos (zu) \mathrm{d}u$ and $\phi$ is some exponentially decaying function.

However, Tao did not provide any reference for this result. So does anyone know where i can find this result ? Which paper of de Bruijn and which lemma/theorem/corollary ?

**ADDENDUM 1:** In the same video (min 37-38), Tao states an even more interesting result of De Bruijn:

``*De Bruijn showed that if for some real $t_0$ the zeros of $H_{t_0}$ are contained in the horizontal strip $\lbrace x+iy: |y|\leq y_0 \rbrace$, then for all $t>t_0$, they will be contained in a narrower strip \begin{equation} \lbrace x+iy: |y|\leq (y_{0}^2 - 2(t-t_0))^{1/2}_{+} \rbrace,\end{equation} where $x,y \in \mathbb{R}$*.''

Recall that $H_{0}(z)=\frac{1}{8}\xi(\frac{1}{2}+\frac{iz}{2})$, where $\xi$ denotes the Riemann xi function. Since $\xi(s)\neq 0$ for $\Re(s)\geq 1$, we can take $t_0 = 0, y_0 = 1, t=t_{0} + \frac{y_{0}^2}{2} - \epsilon$ where $\epsilon$ is an arbitrarily small positive number, and deduce that all the zeros of $H_t$ must be real for $t>0$. This implies that if all zeros of $H_t$ are real if and only if $t\geq \Lambda$, then $\Lambda \leq 0$ (the Riemann Hypothesis !)

In other words, **De Bruijn proved the RH, but didn't realize it !** This strengthens the need for a complete reference of de Bruijn's results that Tao quoted.

**ADDENDUM 2:** The de Bruijn result only entails that $\Lambda\leq 1/2$, as explained by @Conrad in the comments.