Stability and hyperbolicity

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7. Riemann $\Xi$ -function

Let

$\Xi(z):=\int_{-\infty}^{\infty} \Phi(t)\cos(zt) dt,$ where

$\Phi(t):=\sum_{n=1}^\infty (2n^4\pi^2e^{9t}-3n^2\pi e^{5t})\exp(-n^2\pi e^{4t}).$

Let $p_k(z)$ be the polynomials orthogonal with respect to $\Phi$ on $(-\infty,\infty)$ . Define $P_n(z):=\frac{\det\left[p_i(z_j)\right]_{i,j=1}^n}{\det\left[z_i^{j-1}\right]_{i=1,j=1}^n}, \qquad\text{and}\qquad Q_n(z):=\frac{\det\left[p_i(z_j)\right]_{i,j=2}^n}{\det\left[z_{i}^{j-2}\right]_{i,j=2}^{n}}.$

Add remark

Problem 7.1.
[D. Dimitrov] Investigate the zeros of $P_n$ and $Q_n.$
The motivation for this problem is the following. If for all $n\in\mathbb{N}$ , $P_n(z,z,\ldots,z)$ vanishes only when $z=i\alpha$ , with $\alpha\in\mathbb{R}$ , then the Riemann hypothesis is true. Alternatively, if for each $n\in\mathbb{N}$ , $Q_n(z,z,\ldots,z)$ is non-zero on the imaginary axis, then the Riemann hypothesis is true.
1. Remark. [D. Dimitrov] The following generalization of the result mentioned above is true.
  
  Let $\mu$ be an even, positive measure on $\mathbb{R}$ , and $F(z):=\int_{-\infty}^\infty\cos(zt)d\mu(t)$ . Let $\{p_k(z)\}_{k=0}^\infty$ be the orthogonal polynomials with respect to $\mu$ on $(-\infty,\infty)$ . Then $F\in\mathcal{L}\text{-}\mathcal{P}$ if and only if the associated polynomial $P_n[p_1,\ldots,p_n]$ has purely imaginary zeros, if and only if $Q_n[p_1,\ldots, p_n]$ has no purely imaginary zeros. (for all $n$ )
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Problem 7.2.
[J. Haglund] Are there any non-real zeros of $\int_{0}^\infty \Phi^\alpha(t)\cos(zt)dt \qquad \text{for}\qquad \alpha>0 ?$

Cite this as: AimPL: Stability and hyperbolicity, available at http://aimpl.org/hyperbolicpoly.