# P12929: Optimal Gaussian Rank-One Loewner Threshold for the Riemann Theta Kernel

- ID: `P12929`
- Reference: `optimal-gaussian-rank-one-loewner-threshold-riemann-theta-kernel`
- Page: https://theoremdb.org/statements/P12929
- 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/optimal-gaussian-rank-one-loewner-threshold-riemann-theta-kernel?representation=markdown
- Record maturity: Reviewed problem

## The problem

Let \(\Phi\) be the classical even Riemann theta kernel appearing in the Fourier representation
\[
\Xi(z)=2\int_0^\infty \Phi(t)\cos(zt)\,dt.
\]
For real \(a,b\), define
\[
C_\Phi(a,b)=2\int_{|a+b|/2}^{\infty}q\,\Phi\!\left(\frac{a-b}{2}+q\right)\Phi\!\left(\frac{a-b}{2}-q\right)\,dq.
\]
Write \(D(a)=C_\Phi(a,a)\), and define the normalized kernel
\[
R_\Phi(a,b)=\frac{C_\Phi(a,b)}{\sqrt{D(a)D(b)}}.
\]
The diagonal quantity \(D(a)\) is positive. Fix the Gaussian function
\[
u(a)=e^{-3a^2/10}.
\]
Among all nonnegative real numbers \(\beta\) for which the kernel
\[
R_\Phi(a,b)-\beta\,u(a)u(b)
\]
is positive semidefinite on \(\mathbb R\), let \(\beta_*\) denote the supremum; if there is no positive admissible \(\beta\), set \(\beta_*=0\).

Determine whether
\[
\beta_*\ge \frac{19}{20}.
\]
Equivalently, prove or disprove that for every integer \(n\ge1\) and every choice of real numbers \(a_1,\dots,a_n\), the matrix
\[
\left[R_\Phi(a_i,a_j)-\frac{19}{20}e^{-3a_i^2/10}e^{-3a_j^2/10}\right]_{i,j=1}^n
\]
is positive semidefinite.

A positive solution must come directly from the theta-side structure above. It may not assume the Riemann hypothesis, the reality of the zeros of \(\Xi\), Hermite--Biehler or de Branges positivity, the Schur quotient \((\Xi-i\Xi')/(\Xi+i\Xi')\), or a pre-existing positivity theorem for \(C_\Phi\) or \(R_\Phi\). A negative solution may consist of an explicit finite configuration, a local jet obstruction, an asymptotic obstruction, or another rigorous certificate. A stronger solution may determine \(\beta_*\) exactly or establish rigorous bounds that decide the stated inequality.

## Status

**Resolution status:** Open.

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

## Research packet

### Working on this

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## References

No external mathematical reference has been recorded for this problem.