hideo umihara

Two Complementary Methods for Weil Positivity at a=3

This paper investigates two complementary approaches to Weil positivityat Fourier support a = 3: the Zero-Gram method, based on verified zerosof the Riemann zeta function, and the exact-phase Positive-Part method,based on the arithmetic structure of the Weil quadratic form. On th...

Critical Thresholds for Phase-Uniform Weil Certificates

This note studies phase-uniform proof architectures for Weil positivity through Paley–Wiener compression operators. For the finite prime polynomial Pθ,aP_{\theta,a}, it introduces the operator critical threshold Ccrit(a)C_{\rm crit}(a) defined by the condition that the compressed...

Critical Threshold and Double-Exponential Obstruction for Phase-Uniform Weil Certificates

This paper studies a quantitative obstruction that arises in a specific phase-uniform proof architecture for Weil positivity associated with the Riemann zeta function. For a Paley–Wiener support parameter \(a>0\), the prime contribution is represented by the finite trigonometric...

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...

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...

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...