10. Weighted prime halfspaces
STATUS: RIEMANN HYPOTHESIS OPEN
Recorded 2026-09-30. Date of the recorded report; individual start and finish times are not reconstructed.
Starting idea
Write squarefree integers as prime subsets subject to a logarithmic weight bound. The question was whether this structure forces cancellation between even and odd subset sizes.
What was tested
The track checked finite-difference bounds, exact large-prime decompositions, Gibbs tilts and the Selberg–Delange family S_z(X) = Σ_{n≤X} μ²(n)z^{ω(n)}. Here μ is the Möbius function and ω counts distinct prime factors.
What survived
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The finite combinatorial identities preserve parent multiplicities and the actual signs. Generic central-binomial bounds can be sharp even with rationally independent weights.
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At z = −1, S_z is M(X). All local Selberg–Delange asymptotic coefficients vanish through reciprocal-Gamma factors; this is a statement about an asymptotic expansion.
What failed or remains unproved
Vanishing local coefficients is not a finite-X pairing and does not bound the remaining global term. Unsigned influence or anti-concentration estimates do not evaluate the needed signed covariance. Tilt reconstruction is exact only when its normalization and covariance terms are retained.
Why the work moved on
The next track isolated the global remainder and asked what a smoothed explicit formula actually determines.
Current scope
Local expansions and exact finite identities retained; no fixed-power cancellation estimate obtained.
“Proved” here refers to the stated mathematical claim in the linked research record. Internal AI-agent checks are scoped checks, not external peer review. Numerical and formal checks have separate limits described in Reproducibility. No priority or novelty claim is made.
Canonical records
- Detailed source report
- Completion report
- Scoped internal audit
- Historical state record
- Recorded validation
The source archive preserves the original Japanese research prose. These links point to publication copies, not third-party paper downloads. The source map explains historical state files and references distinguish publication status from verification.