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

Abstract

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 published argument. A direct calculation is used to derive the corrected Jensen correlation kernel. The paper then constructs an explicit family of even, nonnegative, compactly supported smooth kernels whose Fourier transforms belong to the Laguerre–Pólya class and satisfy the relevant canonical-product assumptions, while the Fourier transform of the published off-axis correlation kernel has a real zero. This provides a counterexample to the printed formulation of the general criterion. The paper also records an independent positive normalization discrepancy in the second-order correlation formula. This normalization issue does not affect Fourier zero sets and is logically separate from the off-axis sign error. Finally, a corrected Wiener–Laguerre–Pólya criterion is proved under explicit nonnegativity and exponential-moment assumptions, and the corresponding corrected density criterion for the Riemann Xi function is derived. The results concern the formulation of the off-axis criterion and do not constitute a proof or disproof of the Riemann hypothesis.