Determine whether lambda^(L)≥ 0 for every L>0, where lambda^(L)=infQ(f)/lVert f rVert_2^2:0≠ f, operatornamesuppf⊆[-L,L] and Q is Weil’s quadratic form. Equivalently, determine whether Q(f)≥ 0 for every admissible compactly supported test function f; by Weil’s criterion, this is equivalent to the Riemann Hypothesis.

Riemann Hypothesis via Weil’s positivity criterion

Determine whether $\lambda^*(L)\ge 0$ for every $L>0$ , where $\lambda^*(L)=\inf\left\{Q(f)/\lVert f\rVert_2^2:0\ne f,\ \operatorname{supp}f\subseteq[-L,L]\right\}$ and $Q$ is Weil’s quadratic form. Equivalently, determine whether $Q(f)\ge 0$ for every admissible compactly supported test function $f$ ; by Weil’s criterion, this is equivalent to the Riemann Hypothesis.

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

Weil’s criterion for the Riemann Hypothesis
The Riemann Hypothesis holds if and only if $Q(f)\ge 0$ for every admissible compactly supported test function $f$ , where $Q$ is Weil’s quadratic form.
source: Weil positivity in compact windows: certified two-sided bounds and a Landau--Widom decay law

References

Primary source

arXiv

Additional references

Weil positivity in compact windows: certified two-sided bounds and a Landau--Widom decay law — arXiv

analytic-number-theory computational-mathematics riemann-hypothesis

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed progress

A new finite-window calculation extends a known positivity check, but it does not prove the Riemann Hypothesis.

The problem studies whether checking Weil’s positivity criterion on increasingly large finite windows can establish the Riemann Hypothesis. The latest report gives an unconditional certificate at $L=0.8$ , but explicitly treats it as a special case rather than a global proof.

August 25, 2026 finite-window certificate

The paper reports an unconditional certificate at $L=0.8$ , unconditional upper bounds for larger windows, and an asymptotic bound conditional on $\mathrm{RH}$ . These claims extend the cited finite-window range but have no independent verification in the supplied evidence. Related work attributed to Claude improves a lower bound for zeros on the critical line, not $\mathrm{RH}$ itself, and does not verify this certificate.

Current status (as of August 2026): The reported $L=0.8$ certificate and larger-window bounds are claimed but unverified; no proof of $\mathrm{RH}$ or independently confirmed resolution of the finite-window problem has been found.

Sources

No solutions have been posted yet.

Weil’s criterion for the Riemann Hypothesis

Primary source

Additional references

Progress summary