Akihiro Koide

High-Order Center Failures of the Derivative Laguerre Inequalities for the Riemann Xi Kernel

This preprint studies the high-order behavior of the derivative Laguerre inequalities for the standard Jacobi-theta kernel associated with the Riemann Xi function. The paper focuses on the symmetry point at the origin and derives a three-term asymptotic expansion for the even der...

A Counterexample to Csordas' Open Problem 4.13 for the Riemann Xi Kernel

This preprint gives a negative answer to Csordas’ Open Problem 4.13 concerning derivative Laguerre inequalities for the Jacobi theta kernel associated with the Riemann Xi function. At the symmetry point, the proposed inequalities are proved to hold through derivative order eight...

The First Failure of the Derivative Laguerre Inequalities for the Riemann Xi Kernel

This preprint studies the derivative Laguerre inequalities for the standard kernel associated with the Riemann Xi function, in connection with Csordas’ Open Problem 4.13. The paper proves that the inequalities hold globally for the first eight derivative shifts. It then shows tha...

Arithmetic of CM Center Polynomials for the Riemann Xi Kernel

This preprint studies the arithmetic structure of the even derivatives at the symmetry center of the standard Jacobi-theta kernel associated with the Riemann Xi function. Using Romik’s centered expansion of the Jacobi theta constant, the paper constructs a family of rational poly...

Unbounded Sign-Change Excess in Derivatives of the Riemann–Jacobi Kernel

This preprint studies the real-zero geometry of the derivatives of the standard Riemann–Jacobi kernel associated with the Riemann Xi function. The main result shows that every failure of the derivative Laguerre inequality at the symmetry point forces two additional sign changes b...

Unbounded Sign-Change Excess in Derivatives of the Riemann–Jacobi Kernel

This preprint studies the real-zero geometry of the derivatives of the standard Riemann–Jacobi kernel associated with the Riemann Xi function. The main result shows that every failure of the derivative Laguerre inequality at the symmetry point forces two additional sign changes b...

A Counterexample to Csordas' Open Problem 4.13 for the Riemann Xi Kernel

This preprint gives a negative answer to Csordas’ Open Problem 4.13 concerning derivative Laguerre inequalities for the Jacobi theta kernel associated with the Riemann Xi function. At the symmetry point, the proposed inequalities are proved to hold through derivative order eight...

A Counterexample to an Off-Axis Wiener Criterion for the Laguerre–Pólya Class and a Corrected Jensen Correlation Kernel

This preprint examines an off-axis Wiener criterion for the Laguerre–Pólya class proposed by Dimitrov and Xu. It identifies a missing complex factor in the Fourier transform of an odd kernel, which reverses the sign of one of the second-order correlation terms used in the publish...

Exact Order-Lifting Depth and Ramification in Global Function Fields: A Coefficient-Field Rank Formula and Counterexamples to Kernel-Dimension Conjectures

This preprint studies order lifting for rational functions on curves over finite fields. It identifies order-lifting depth with the local multiplicity of the associated morphism and relates this invariant to Fermat quotients, Hasse derivatives, and the lifting of multiplicative o...

Exact Order-Lifting Depth and Ramification in Global Function Fields: A Coefficient-Field Rank Formula and Counterexamples to Kernel-Dimension Conjectures

This preprint studies order lifting for rational functions on curves over finite fields. It identifies order-lifting depth with the local multiplicity of the associated morphism and relates this invariant to Fermat quotients, Hasse derivatives, and the lifting of multiplicative o...