Zeros of the derivative of Riemann's $\xi$-function

The Riemann hypothesis implies that the zeros of derivatives of all orders of $\xi$ lie on the critical line.

B. Conrey, Zeros of derivatives of Riemann’s xi-function on the critical line, J. Number Theory 16 (1983), 49-74.

In exercise 1 on page 443 of their book "Multiplicative Number Theory," Montgomery & Vaughan outline a proof of the statement:

"Assuming the Riemann Hypothesis, $\xi'(s)=0 \implies \mathrm{Re}(s)=1/2$."

Assuming RH, let $s=\sigma+it$ and let $\rho=\frac{1}{2}+i\gamma$ denote a zero of $\xi(s)$. The main idea of their argument is that, on RH, it follows from that Hadamard product for $\xi(s)$ that $$ \mathrm{Re} \frac{\xi'}{\xi}(s) = \sum_{\rho} \mathrm{Re}\frac{1}{s-\rho} = \sum_\gamma \frac{\sigma-1/2}{(\sigma-1/2)^2+(t-\gamma)^2}.$$ Now if $\xi'(s)=0$, then the left-hand side of the above expression is zero. On the other hand, the only way that the sum over $\gamma$ vanishes is if $\sigma=1/2$, i.e. $\mathrm{Re}(s)=1/2$.

The Riemann hypothesis implies that the function $\Xi(z)=\xi(1/2+iz)$ is in the Laguerre-Pólya class. Therefore it is a limit, uniformly on compact sets, of a sequence of polynomials with real roots. The derivatives are also again in the same class and have therefore only real zeros. It follows that all zeros of $\xi(s)$ and its derivatives will be on the critical line.

I imagine that this is due to Pólya, but have not his Collected Works to confirm.

Given the simplicity of the proof I noticed, I will give a sketch of it. We have $$\Xi(t)=\Xi(0)\prod_{n=1}^\infty \Bigl(1-\frac{t^2}{\alpha_n^2}\Bigr).$$ The Riemann hypothesis is that all $\alpha_n$ are real.

Therefore, assuming RH, $\Xi(t)$ is the limit uniformly in compact sets of $\bf C$ of the polynomials $$P_N(t):=\Xi(0)\prod_{n=1}^N \Bigl(1-\frac{t^2}{\alpha_n^2}\Bigr).$$ We are assuming that all roots of these polynomials are real. Therefore the same will happen with any derivative $P_N^{(k)}(t)$.

By the general Theorems of Complex Analysis $\lim_{N\to\infty}P_N^{(k)}(t)=\Xi^{(k)}(t)$ uniformly in compact sets. By the argument principle any zero of $\Xi^{(k)}(t)$ is limit of zeros of $P_N^{(k)}(t)$. Therefore any zero of $\Xi^{(k)}(t)$ is real. The relation $\Xi(t)=\xi(\frac12+it)$ implies that all the zeros of the derivatives of $\xi(s)$ are in the critical line.

(It is essentially contained in the paper by G. Pólya, Bemerkung zur Theorie der ganzen Funktionen, Collected papers II, 154--162.)