# Reference request for some result of de Bruijn on zeros of some holomorphic function

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 $$$$\lbrace x+iy: |y|\leq (y_{0}^2 - 2(t-t_0))^{1/2}_{+} \rbrace,$$$$ where $$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.

• Thanks Todd Trimble for including the video link. – radian Jul 9 at 9:17
• You get $t \ge .5$ since $y_0=1$, so you just noticed Bruijn's conclusion that $\Lambda \le .5$ – Conrad Jul 9 at 13:22
• 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 $\theta(x)$ – reuns Jul 9 at 13:26
• 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) – Conrad Jul 9 at 13:56
• @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. – radian Jul 9 at 23:17