RIEMANN RESEARCH RECORD · NS-53 · 22 SEPTEMBER 2026 · V1

An exact block formula, with the arithmetic gain still open

The approximation route needs convergence at any rate. This checkpoint identifies the exact improvement from adding integer dilations and the arithmetic estimate still missing.

RH and G2 remain open. The results below are exact identities and conditional implications; there is no new numerical certificate or convergence claim.

The correction to the earlier assessment

In the strong Nyman–Beurling criterion, the squared distance E_N from χ = 1_(0,1) to the span of ρ_n(x) = {1/(nx)}, n ≤ N, need only tend to zero. A bound such as E_N = O((log N)^−1/2) would suffice. The known lower bound does not require proving a matching sharp upper rate.

What adding a block actually buys

Let G describe the old vectors, D the new block, and C their cross terms. Let b and b_B be their correlations with χ. The old optimal coefficients are c = G⁻¹b. Two corrections are essential:

These formulas project the entire new block off the old span. They retain the cross terms and Gram conditioning. The derivation is standard Hilbert-space geometry specialized to the arithmetic dictionary, not a new estimate towards RH.

For κ = log(2π) − γ, an unconditional lower bound follows:

E_N − E_M ≥ ‖h‖² / [κ log(M/N)].

For a block N < n ≤ qN, the denominator is bounded independently of N. The unknown is the numerator: how much correlation the actual optimal residual retains with the new arithmetic block.

An elementary arithmetic checkpoint

The proof gives exact integral and convergent cell-series expressions for every Gram entry. It also expresses the whole residual through divisor sums on each interval j ≤ 1/x < j+1.

For the unsmoothed Möbius choice c_n = −μ(n), the first N+1 cells alone contribute

(N+1) |∑_n≤N μ(n)/n|².

This is a necessary penalty for that particular coefficient choice, not a lower bound on the optimal distance. The raw Möbius approximation is already known to fail in this norm; the checkpoint does not claim a new obstruction.

The open input

A sufficient residual block-correlation hypothesis is written explicitly in the proof. At N_j = q^j, it asks for ‖h_j‖² ≥ κ log q · α_jE_Nj, where 0 ≤ α_j < 1 and ∑α_j diverges. The hypothetical choice α_j = 1/[2(j+1)] would already imply the slower rate above.

No such asymptotic estimate is proved. Nor is this sufficient hypothesis asserted to follow from RH. The trace bound may discard large gains in directions associated with small eigenvalues of S. An exact block quotient could succeed where the trace estimate fails.

The finding is a missing estimate. It establishes neither failure of the stated sufficient hypothesis nor failure of the approximation object. More finite distances would not settle it. No new Weil window or tail metric was introduced.

Primary sources and verification

Báez-Duarte, Theorem 1.1 and Introduction establishes the qualitative criterion and records the raw-prefix failure. Burnol, Advances in Mathematics 170 (2002) supplies the lower-bound result. Existing numerical work is described by Landreau–Richard, Experimental Mathematics 11 (2002).

The accompanying proof contains complete derivations of the new checkpoint’s identities and conditional implication. Manuscript integration, build page count, and reference counts are recorded by the integrating author; this document does not claim that a source check is a manuscript build.

Claim categories: exact identity · proved implication · named open input. G2 and RH remain open.