On some positivity properties of the Riemann $\Xi$ function
We prove strict log-concavity of the positive functions $\Theta$ and $\Lambda$ with one-sided and bilateral Laplace transforms $1/\xi(\tfrac12+\sqrt s)$ and $1/\Xi$, respectively, where $\Xi(z)=\xi(\tfrac12+iz)$. For either function, log-concavity is the Pólya frequency property of order two (PF$_2$). Without assuming RH, but using known zero verification through height $3\cdot10^{12}$, we further prove that $\Lambda$ is a PF function of order ten (PF$_{10}$). We also prove local PF positivity for $\Theta$: for each fixed order, the ordered minors of $\Theta(x-y)$ are positive when all…
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