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7. Riemann $\Xi$-function

Let \[\Xi(z):=\int_{-\infty}^{\infty} \Phi(t)\cos(zt) dt, \] where \[\Phi(t):=\sum_{n=1}^\infty (2n^4\pi^2e^{9t}-3n^2\pi e^{5t})\exp(-n^2\pi e^{4t}).\]

    1.     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 $n$)


            • Add remark

              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.