# P12931: First Even Dirichlet–Schur Pivot Asymptotics for the Riemann Theta Kernel

- ID: `P12931`
- Reference: `first-even-dirichlet-schur-pivot-asymptotics-riemann-theta-kernel`
- Page: https://theoremdb.org/statements/P12931
- 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/first-even-dirichlet-schur-pivot-asymptotics-riemann-theta-kernel?representation=markdown
- Record maturity: Reviewed problem

## The problem

Let \(\Phi\) be the classical even Riemann theta kernel and define
\[
C_\Phi(a,b)=2\int_{|a+b|/2}^{\infty}q\,\Phi\!\left(\frac{a-b}{2}+q\right)\Phi\!\left(\frac{a-b}{2}-q\right)\,dq.
\]
Fix the interval \([-2,2]\). For each even integer \(m\ge 2\), set \(h=4/m\) and \(t_j=-2+jh\) for \(0\le j\le m\), and form the anchor Gram matrix \(G_m=[C_\Phi(t_i,t_j)]_{0\le i,j\le m}\). Whenever \(G_m\) is invertible, define
\[
c_m(a)=\bigl(C_\Phi(t_0,a),\ldots,C_\Phi(t_m,a)\bigr)^T
\]
and the intrinsic Schur residual
\[
S_m(a,b)=C_\Phi(a,b)-c_m(a)^*G_m^{-1}c_m(b).
\]
For the cell \(I_j=[t_j,t_{j+1}]\), define the normalized Dirichlet modes
\[
\phi_{j,k}(x)=\sqrt{\frac2h}\sin\!\left(k\pi\frac{x-t_j}{h}\right),\qquad k\ge1.
\]
For \(k,\ell\ge1\), let \(H_{k\ell}^{(m)}\) be the \(m\times m\) block whose \((i,j)\) entry is
\[
(H_{k\ell}^{(m)})_{ij}=\int_{I_i}\int_{I_j}S_m(x,y)\phi_{i,k}(x)\phi_{j,\ell}(y)\,dx\,dy.
\]
Whenever \(H_{11}^{(m)}\) is invertible, define
\[
P_1^{(m)}=H_{11}^{(m)},\qquad
P_2^{(m)}=H_{22}^{(m)}-H_{21}^{(m)}(H_{11}^{(m)})^{-1}H_{12}^{(m)}.
\]
Determine the exact small-cell asymptotic of the first even modal pivot \(P_2^{(m)}\) as \(m\to\infty\), equivalently \(h\downarrow0\). In particular, decide whether there exist constants \(m_0\) and \(0<c_-\le c_+<\infty\) such that for every even \(m\ge m_0\), the matrices \(G_m\) and \(H_{11}^{(m)}\) are invertible, the pivots \(P_1^{(m)}\) and \(P_2^{(m)}\) are positive semidefinite, and
\[
c_-h^6\le \frac{\operatorname{Tr}P_2^{(m)}}{\operatorname{Tr}P_1^{(m)}}\le c_+h^6.
\]
Preferably determine whether the finite positive limit
\[
\lim_{m\to\infty}h^{-6}\frac{\operatorname{Tr}P_2^{(m)}}{\operatorname{Tr}P_1^{(m)}}
\]
exists and identify it. A negative resolution may instead exhibit a rigorous infinite sequence of even \(m\to\infty\) for which the required invertibility, positivity, or either side of the \(h^6\) comparison fails.

## 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.