[#P12929] Optimal Gaussian Rank-One Loewner Threshold for the Riemann Theta Kernel
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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.
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2See also
- Improve the upper exponent for the longest Pierce remainder chainanalytic number theory
- Riemann hypothesisanalytic number theory
- Cramér's prime-gap conjectureanalytic number theory
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“Optimal Gaussian Rank-One Loewner Threshold for the Riemann Theta Kernel.” TheoremDB. P12929. Problem statement; statement identity tdbc1:84a0f794d747e851a88c6b6f7fa8eb024228e7408f7c0abeeab6cc62a0a25ea2; statement text SHA-256 316ff6f4dc7dbc8edf607f070f6010ecb15cd1d0ff0b5bc254e73bd8bb3ae5a5. https://theoremdb.org/statement/?ref=P12929
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title = {{Optimal Gaussian Rank-One Loewner Threshold for the Riemann Theta Kernel}},
howpublished = {TheoremDB},
note = {Problem statement; statement identity tdbc1:84a0f794d747e851a88c6b6f7fa8eb024228e7408f7c0abeeab6cc62a0a25ea2; statement text SHA-256 316ff6f4dc7dbc8edf607f070f6010ecb15cd1d0ff0b5bc254e73bd8bb3ae5a5},
url = {https://theoremdb.org/statement/?ref=P12929}
}Plain text: Built Markdown snapshot
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