Zeros of the derivative of $\xi$

Question

In his paper on Zeros of derivatives of Riemann $\xi$-function on critical line Brian Conrey mention that

It can be shown that the Riemann hypothesis implies that all zeros of $\xi (s)$, the derivative of $\xi$-function, have real part $1/2$ for any $m$. Where $$\xi(s)=\frac{1}{2}s(s-1)\pi^{-s/2}\Gamma\bigg(\frac{s}{2}\bigg) \zeta(s).$$

Could anyone kindly recommend some references, books, or scholarly articles that elaborate on this topic? I would greatly appreciate your assistance. Thank you in anticipation!

Answer 1

Assuming the Riemann hypothesis, the $\xi(1/2+iz)$-function belongs to the Laguerre-Polya class (defined as the closure of polynomials with all zeros real). This follows from the parametric description of this class obtained by Laguerre and Polya: it consists exactly of the functions of the form $$cz^me^{-az^2+bz}\prod\left(1-\frac{z}{a_k}\right)e^{z/a_k},$$ where $a\leq 0,$ $b$ and $a_k$ are real. Since Riemann $\zeta$ and $\xi$ have order $1$, Hadamard's factorization implies that $\zeta(1/2+iz)$ has this form (assuming that all $a_k$ are real), thus it belongs to the Laguerre-Polya class, and then it evident from the definition this class, that all zeros of all derivatives are also real.