On some positivity properties of the Riemann $Ξ$ function

Abstract

We prove strict log-concavity of the positive functions $Θ$ and $Λ$ with one-sided and bilateral Laplace transforms $1/ξ(\tfrac12+\sqrt s)$ and $1/Ξ$, respectively, where $Ξ(z)=ξ(\tfrac12+iz)$. For either function, log-concavity is the Pólya frequency property of order two (PF$_2$). Without assuming the Riemann hypothesis (RH), but using its published verification up to height about $3\cdot10^{12}$, we further prove that both densities are PF$_{k_*}$ for $k_*=1\,810\,540\,931\,075$. The proof views $Θ$ as a small signed perturbation of a density that is PF of every order, with an error bound independent of the order, so the numerical check reduces to scalar inequalities. Since $Λ$ is a Gaussian mixture of $Θ$, it inherits the same order. Conversely, a pole obstruction shows that if either density is PF$_k$, then every zeta zero lower than the $(k+2)$-nd lies on the critical line. Finally, we prove that if RH holds up to height $T>\tfrac12$, then $Ξ$ satisfies the generalized Laguerre inequalities $L_n[Ξ](x)\ge0$ of every order $n$ for $|x|\le T-\tfrac12$, extending the case $n=1$ of Csordas, Ruttan and Varga (1991).