Incorporate the imaginary-part identity into the matrix method to constrain clustering

Investigate whether the sign-sensitive information supplied by the imaginary-part identity for the smoothly truncated logarithmic derivative of the Riemann zeta-function can be incorporated into the matrix construction underlying the unconditional simple-zero result, with the aim of obtaining stronger restrictions on clustering than those obtained from the trace and squared-trace identities alone.

Background

The paper develops an unconditional matrix argument in which vectors associated with zeta zeros produce a symmetric matrix whose first two trace moments imply that more than two-thirds of the zeros are simple and lie on the critical line. This method overcomes the cancellation caused by clusters of zeros away from the critical line by tracking positive and negative directions rather than relying on termwise positivity.

The newly derived imaginary-part identity has an odd, sign-sensitive kernel. The authors suggest that it may distinguish different arrangements of the same directions and provide information about how positive and negative directions associated with off-line zeros align and cancel. The paper does not develop this extension, leaving open whether the additional phase information can improve the restrictions on clustering or the resulting zero-proportion bounds.