Akihiro Koide

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...

Power-Saving Center-Failure Counts for the Derivative Laguerre Inequalities of the Riemann Xi Kernel

This preprint studies the asymptotic number of failures of the derivative Laguerre inequalities at the symmetry point of the standard Jacobi-theta kernel associated with the Riemann Xi function. Building on a previous transition analysis, it strengthens the known qualitative coun...

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 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...

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...

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...

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...

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...

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...