math.CA — Classical Analysis and ODEs

On some positivity properties of the Riemann $\Xi$ function

Contributed by Suvrit Sra

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 differences $x_i-y_j$ are sufficiently small and positive. Additionally, we prove the generalized Laguerre inequalities: if RH holds up to height $T>1/2$, then $L_n[\Xi](x)\ge0$ for every $n\ge0$ and $|x|\le T-1/2$; the case $n=1$ is due to Csordas, Ruttan and Varga (1991).