Rule out cancellation of off-line and critical-line zero interactions

Prove that configurations of off-line zeros and critical-line zeros cannot produce sufficient negative mixed interactions to cancel the positive contribution of off-line zeros and thereby obstruct a direct upper bound for the second moment of zeta-zero multiplicities in the unconditional scalar pair-correlation identity.

Background

The scalar unconditional pair-correlation identity contains diagonal terms involving the squared multiplicities of zeta zeros and off-diagonal interactions governed by the complex Fejér kernel. Interactions among off-line zeros have a positive aggregate contribution, but interactions between reflected off-line pairs and nearby critical-line zeros can be negative.

The paper explains that clustering and multiplicity may amplify these negative mixed interactions. A direct extension of Montgomery’s positivity argument would require proving that the negative terms cannot compensate for the excess diagonal mass caused by multiple zeros. The authors explicitly state that they were unable to rule out such configurations, although the matrix method handles the obstruction by retaining spectral information.

References

In the unconditional scalar argument, we were not able to rule out configurations in which the accumulation of such negative mixed terms cancels enough of the positive contribution coming from the off-line zeros to interfere with Montgomery's multiplicity argument. Consequently, unlike under the Riemann Hypothesis, one cannot simply discard the off-diagonal terms and obtain an upper bound for the second moment of the multiplicities.

— On the Imaginary Part of $ζ'/ζ$  (2610.12853 - Aryan, 9 Oct 2026) in Appendix, Section “An Example of the Clustering Obstruction”