hideo umihara

Interlacing and Derivative Log-Concavity for the Riemann–Jacobi Kernel

This paper gives rigorous new proofs for the unresolved Coffey–Csordas Conjecture 2.5 concerning the derivative Turán inequalities of the Riemann–Jacobi kernel. Specifically, it proves the previously unresolved cases [n=2,3,4,5,] in addition to the known case (n=1). A central fea...

Interlacing and Derivative Log-Concavity for the Riemann–Jacobi Kernel

This paper gives rigorous new proofs for the unresolved Coffey–Csordas Conjecture 2.5 concerning the derivative Turán inequalities of the Riemann–Jacobi kernel. Specifically, it proves the previously unresolved cases [n=2,3,4,5,] in addition to the known case (n=1). A central fea...

A Fourth-Logarithmic-Derivative Inequality for the Riemann Xi Kernel

We prove a global fourth-logarithmic-derivative inequality for the positive even Riemann Xi kernel \[\Phi(x)=\sum_{n=1}^{\infty}\pi n^2\left(2\pi n^2e^{4x}-3\right)\exp\left(5x-\pi n^2e^{4x}\right),\] extended evenly to the real axis. Writing \[L(x)=\log\Phi(x),\] the main result...

A Fourth-Logarithmic-Derivative Inequality for the Riemann Xi Kernel

We prove a global fourth-logarithmic-derivative inequality for the positive even Riemann Xi kernel \[\Phi(x)=\sum_{n=1}^{\infty}\pi n^2\left(2\pi n^2e^{4x}-3\right)\exp\left(5x-\pi n^2e^{4x}\right),\] extended evenly to the real axis. Writing \[L(x)=\log\Phi(x),\] the main result...

Strict Total Positivity of Order Four for the de Bruijn--Newman Kernel

We prove that the de Bruijn--Newman kernel associated with the Riemann Xi function is strictly totally positive of order four. Let \[K(x,y)=\Phi(x-y),\] where \(\Phi\) is the classical positive even Riemann kernel. We establish that for every \[1\le m\le4,\] and every pair of str...

Strict Total Positivity of Order Four for the de Bruijn--Newman Kernel

We prove that the de Bruijn--Newman kernel associated with the Riemann Xi function is strictly totally positive of order four. Let \[K(x,y)=\Phi(x-y),\] where \(\Phi\) is the classical positive even Riemann kernel. We establish that for every \[1\le m\le4,\] and every pair of str...