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Published April 28, 2026 | Version v707
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V120_2 — Zero Packets and Prime Residual Correlations in a Fejér‑Centered Boundary Gap

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Description: This paper investigates the Fejér‑centered boundary gap energy G(a), which bridges two equivalent criteria for the Riemann Hypothesis (RH): the Fejér first‑moment condition and the critical logarithmic mean‑square condition. It identifies the gap as the meeting point of three obstructions: weighted zero‑packet energy, damped zero pair‑correlation, and prime residual long‑memory variance.

🔹 Fejér‑Centered Gap

  • Defined as G(a)=a∫∣S(X)−aB(X)∣2e−2aXdX.

  • Spectral form involves Hardy energy and the Laplace‑Fourier transform of H0(s).

  • Weight function Wa(t) peaks at microscopic boundary scale ∣t∣≈a.

🔹 Zero‑Packet Analysis

  • Each critical zero contributes a packet of the form 1/(a+i(t−y)).

  • Diagonal packet weight: harmless, scales like y−2, summable over zeros.

  • Off‑diagonal correlations: main obstruction; require cancellation or Bessel condition.

  • Introduces damped pair‑correlation kernel Pa(u)=2a2/(4a2+u2).

🔹 Pair‑Correlation Model

  • In dyadic blocks of zeros, packet interactions modeled by Pa(u).

  • A no‑excessive‑clustering bound on zero pairs would imply boundedness of G(a).

  • Links to Montgomery’s pair correlation but with damping.

🔹 Prime‑Side Interpretation

  • Residual measure dM(u)=∑A(n)/n dlog⁡n−eu/2du.

  • Kernel Ka(u,v)=e−2amax⁡(u,v)(1−a∣u−v∣)/4.

  • Governs long‑memory correlations of weighted prime powers.

  • Equivalent to variance of multiplicative interval residuals E(X,H).

🔹 Obstruction Identified

  • On the zero side: off‑diagonal packet correlations and off‑line poles.

  • On the prime side: critical cancellation in residual correlations, not raw prime‑pair estimates.

  • Controlling G(a) requires either a zero‑packet correlation bound or a prime residual correlation bound.

🔹 Conclusion V120_2 shows that the Fejér‑centered gap energy is the precise obstruction between Fejér holomorphy and Hardy boundary energy. It unifies three perspectives:

  1. Zero‑packet energy

  2. Damped pair‑correlation

  3. Prime residual variance

The paper does not prove RH unconditionally, but it diagnoses the exact analytic obstacle that must be overcome.

 

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Dates

Issued
2026-03-07

References

  • Titchmarsh, E. C. (1986). The Theory of the Riemann Zeta-function (2nd ed., revised by D. R. Heath-Brown). Oxford University Press. Edwards, H. M. (1974). Riemann's Zeta Function. Dover Publications. Báez-Duarte, L. (2003). A strengthening of the Nyman–Beurling criterion for the Riemann hypothesis. Rendiconti del Circolo Matematico di Palermo, 52(3), 375–380. https://doi.org/10.1007/s12215-003-0007-1 Conrey, J. B. (2003). The Riemann Hypothesis. Notices of the American Mathematical Society, 50(3), 341–353. Ivić, A. (1985). The Riemann Zeta-Function: Theory and Applications. Dover Publications.