V138_2 — The M = 1 Diagonal Packet Barrier in Translated Mean-Square Estimates for the Logarithmic Derivative of the Zeta Function

Description: This paper investigates translated vertical mean‑square estimates for the logarithmic derivative of the Riemann zeta function near the critical line. These estimates arise naturally when shifted‑contour remainders in smoothed Perron formulae are expressed as vertical convolutions of the logarithmic derivative.

The central idea is that each critical‑line zero contributes a local packet with diagonal mass proportional to the inverse of the distance from the critical line. Since there are about T log T zeros in a height interval of length T, the formal diagonal packet scale of the mean square is of order T log T divided by that distance. This corresponds to a loss exponent of M = 1.

The paper emphasizes that this diagonal packet scale is not an unconditional lower bound for the true mean square, because regular terms and off‑diagonal interactions may interfere. Instead, it identifies the natural scale produced by diagonal packet counting. Consequently, any translated mean‑square estimate with M less than 1 cannot be justified by diagonal arguments alone. Achieving such bounds requires additional structure: off‑diagonal cancellation, renormalization, or stronger zero‑distribution input.

The work also formulates Gram matrices for packet interactions, shows how boundary layers affect zero counts, and demonstrates that even finite intervals preserve the same diagonal scale. It clarifies that ordinary Bessel bounds or frame inequalities still yield the M = 1 scale, not an improvement.

The contribution is diagnostic: it identifies the natural M = 1 diagonal packet barrier and explains why an M < 1 input is a genuinely stronger, zero‑sensitive hypothesis. No proof of the Riemann Hypothesis is claimed.