A Zero-Density Bound Near the Critical Line via Cumulant Matching

We derive a zero-density estimate for the Riemann zeta function near the critical line by combining the Selberg Central Limit Theorem, the Euler product cumulant bound, and a width-corrected Hadamard spike formula. Let \(\delta^* = \exp(-C_E \cdot \sigma_T)\) where \(\sigma_T = \sqrt{\tfrac{1}{2}\log\log T}\) and \(C_E\) is the Euler product analyticity radius. We prove:

\[N(\tfrac{1}{2} + \delta^*, T) \leq T^{1+o(1)}.\]

This improves on Ingham's bound \(N(\sigma, T) \leq T^{3(1-\sigma)+\varepsilon}\) at \(\sigma = \tfrac{1}{2} + \delta^*\).

We further show that a Density Lower Bound (DLB) — the assertion that \(\beta_0 > \tfrac{1}{2}\) implies \(N(\tfrac{1}{2}+\delta^*, T) \geq T^c\) for some \(c > 0\) — suffices to derive the Riemann Hypothesis. The DLB is the single remaining ingredient: a Bohr–Landau type polynomial lower bound on the off-line zero count.

The proof chain consists of 20 machine-verified theorems in the proof kernel proof system with zero unresolved obligations.