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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.

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  • $\begingroup$ Thanks Todd Trimble for including the video link. $\endgroup$ – radian Jul 9 at 9:17
  • $\begingroup$ You get $t \ge .5$ since $y_0=1$, so you just noticed Bruijn's conclusion that $\Lambda \le .5$ $\endgroup$ – Conrad Jul 9 at 13:22
  • $\begingroup$ I don't understand the claim but you can't prove the RH from elementary results ($\approx$ not based on the primes) about $\theta(x) = \sum_n e^{-\pi n^2 x}$'s Mellin transform because the RH fails for things like $\zeta(s,1/4)$ (no Euler product) whereas things like $\sum_{m=0}^3 i^m\theta(x+im/4)$ have quite the same properties as $\theta(x)$ $\endgroup$ – reuns Jul 9 at 13:26
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    $\begingroup$ the claim is wrong as noted; it is just an arithmetic mistake, while the true result is (unsurprisingly) the one obtained by De Bruijn in 1950, that $\Lambda \le .5$, $\Lambda$ being what is now called De Bruijn-Newman constant and which needs to be zero for RH to hold (best known upper bound, 0.22 afaik, lower bound zero being very recent) $\endgroup$ – Conrad Jul 9 at 13:56
  • $\begingroup$ @reuns, the fact $y_0 \leq 1$ is a consequence of the Euler product. So the above argument does apply properties of primes in some way. BTW, some experts now believe it is the non-vanishing of $\zeta(s)$ for $\Re(s)>1$ that is more fundamental than the Euler product. So it's a matter of opinion. $\endgroup$ – radian Jul 9 at 23:17

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