# P12927: Theta-to-Pick Gram Factorization for the Riemann Xi Function

- ID: `P12927`
- Reference: `theta-pick-gram-factorization-riemann-xi`
- Page: https://theoremdb.org/statements/P12927
- 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/theta-pick-gram-factorization-riemann-xi?representation=markdown
- Record maturity: Reviewed problem

## The problem

Let \(\xi(s)=\tfrac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) and let \(\Xi(z)=\xi(\tfrac12+iz)\), so \(\Xi\) is a real even entire function. Let \(\Phi\) denote the standard even Riemann theta kernel normalized by
\[
\Xi(z)=2\int_0^\infty \Phi(u)\cos(zu)\,du,
\qquad
\Phi(u)=\sum_{n\ge1}\bigl(2\pi^2n^4e^{9u/2}-3\pi n^2e^{5u/2}\bigr)e^{-\pi n^2e^{2u}}.
\]
For \(z,w\) in the upper half-plane \(\mathbb H=\{z:\operatorname{Im}z>0\}\), define the Hermitian Pick/Bezout kernel
\[
B_\Xi(z,w)=\frac{\Xi(z)\overline{\Xi'(w)}-\Xi'(z)\overline{\Xi(w)}}{z-\overline w}.
\]
Construct, directly from \(\Phi\) and explicit integral transforms, an unconditional positive-kernel representation of the form
\[
B_\Xi(z,w)=\int_\Omega \Psi_z(\omega)\overline{\Psi_w(\omega)}\,d\mu(\omega),\qquad d\mu\ge0,
\]
valid for all \(z,w\in\mathbb H\), with convergence and Hermitian analyticity justified. The construction must not assume the Riemann Hypothesis, zero-freeness of \(\Xi\) in \(\mathbb H\), Hermite--Biehler/de Branges positivity, or the Pick positivity being proved. Prove as part of the solution that such a factorization implies that \(-\Xi'/\Xi\) is a Herglotz/Nevanlinna function on its zero-free domain and forces all zeros of \(\Xi\) to be real, hence proves the Riemann Hypothesis.

A valid negative resolution may instead prove an obstruction theorem for a mathematically natural, explicitly specified class of theta-side factorizations (for example theta-local, finite-order, tensor-product, or finitely generated transforms), showing that no representation in that class can yield the required global positive kernel. Mere failure of a particular ansatz or finite numerical testing does not count.

## 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 `theta-pick-gram-factorization-riemann-xi`, 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.