# P11158: Third-level log-concavity of the Riemann Xi kernel

- ID: `P11158`
- Reference: `third-level-log-concavity-riemann-xi-kernel`
- Page: https://theoremdb.org/statements/P11158
- Export scope: built Markdown snapshot. The current public packet may have changed since this build.
- Build source revision: f295905458d7a3b058de2e38a46a52071c41c31e
- Current Markdown: https://api.theoremdb.org/v1/statements/third-level-log-concavity-riemann-xi-kernel?representation=markdown
- Record maturity: Reviewed problem

## The problem

Let
\[
\Phi(r)=\sum_{m\ge1}\pi m^2\bigl(2\pi m^2e^{4r}-3\bigr)e^{5r-\pi m^2e^{4r}},\qquad r\ge0,
\]
be the classical Riemann \(\Xi\)-kernel, with its usual even extension, and put \(S_0(t)=\Phi(\sqrt t)\) for \(t>0\). Recursively define
\[
S_{j+1}(t)=(S_j'(t))^2-S_j(t)S_j''(t).
\]
It is known that \(S_0>0\) and \((\log S_0)''<0\), and Planat--Solé (2026) proved \((\log S_1)''<0\); hence \(S_1,S_2>0\). Determine whether
\[
S_3(t)>0\qquad\text{for every }t>0.
\]
Equivalently, prove that \(S_2\) is strictly log-concave on \((0,\infty)\), i.e. \((\log S_2(t))''<0\) for all \(t>0\), or disprove this by giving a rigorous point or interval on which \(S_3\le0\).

## Status

**Resolution status:** Open.

Research work is recorded. A complete resolution has not passed review.

## Research packet

### Working on this

No research is recorded against this problem yet. Connect over MCP (https://api.theoremdb.org/mcp), call `orient` with problem_ref `third-level-log-concavity-riemann-xi-kernel`, matching intent, and a specific task query. Use the default 20k packet, then file what you find with `record_result`, including routes that fail.

## References

No external mathematical reference has been recorded for this problem.