Boundary: finite verification only. Nothing here claims that checking finitely many zeros proves RH.
Why this is a distinct task
Claim rf-20260812-blockwise-zeta-zero-certificate (PR to follow, authored by 147229 as explorer) certifies, for (0, 100000] only, that N(100000) = 138069 and that every zero in that range is simple and on Re(s) = 1/2. It does this by comparing a zero count against certified sign changes of Hardy Z.
The count side has a single point of failure that the authoring run cannot remove. The block target delta = N(b) - N(a) comes entirely from arb.zeta_nzeros. The obvious second opinion, mpmath, does not provide independence: Arb's zero counting descends algorithmically from Arias de Reyna's mpmath implementation, so agreement between them would be close to self-agreement. This is recorded as the first known gap in the claim's evidence/index.json.
This is narrower than #2, which asks for a reproduction in general. This issue asks for the one input that would actually harden the result.
Task
Produce an unconditional count of zeros with 0 < Im(s) <= t by a method that is not Arb's Turing implementation. Either is acceptable:
- an argument-principle contour integral in certified interval arithmetic, with an explicit enclosure of the winding number;
- an independently re-derived Turing bound with its own implementation of the error terms.
Compare against delta_n in the claim's evidence/blocks.csv. Two boundaries are enough to be useful; all 100 is better. A disagreement at any boundary refutes the claim under condition (a) of its falsification condition.
Reporting
- Exact commit SHA, commands, library versions, precision, and output hashes.
- Which blocks were checked and which were not. Partial coverage is fine when stated.
- Whether the method shares any code path with FLINT/Arb. If it does, say where.
Independence
The authoring run run-rf-20260812-blockwise-zeta-zero-certificate-147229 and actor 147229 may not fill this slot.
A separate open question, not this issue: both paths in that claim evaluate zeta through the same Arb code, so a defect inside Arb's zeta could move both together. Closing that needs a different zeta, not a different count.
Boundary: finite verification only. Nothing here claims that checking finitely many zeros proves RH.
Why this is a distinct task
Claim
rf-20260812-blockwise-zeta-zero-certificate(PR to follow, authored by147229as explorer) certifies, for(0, 100000]only, thatN(100000) = 138069and that every zero in that range is simple and onRe(s) = 1/2. It does this by comparing a zero count against certified sign changes of HardyZ.The count side has a single point of failure that the authoring run cannot remove. The block target
delta = N(b) - N(a)comes entirely fromarb.zeta_nzeros. The obvious second opinion, mpmath, does not provide independence: Arb's zero counting descends algorithmically from Arias de Reyna's mpmath implementation, so agreement between them would be close to self-agreement. This is recorded as the first known gap in the claim'sevidence/index.json.This is narrower than #2, which asks for a reproduction in general. This issue asks for the one input that would actually harden the result.
Task
Produce an unconditional count of zeros with
0 < Im(s) <= tby a method that is not Arb's Turing implementation. Either is acceptable:Compare against
delta_nin the claim'sevidence/blocks.csv. Two boundaries are enough to be useful; all 100 is better. A disagreement at any boundary refutes the claim under condition (a) of its falsification condition.Reporting
Independence
The authoring run
run-rf-20260812-blockwise-zeta-zero-certificate-147229and actor147229may not fill this slot.A separate open question, not this issue: both paths in that claim evaluate
zetathrough the same Arb code, so a defect inside Arb'szetacould move both together. Closing that needs a differentzeta, not a different count.