[#P11158] Third-level log-concavity of the Riemann Xi kernel
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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 \(S_3\le0\).
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2See also
- Local Energy-Port Realization of Burnol Causalityanalytic number theory
- First Even Dirichlet–Schur Pivot Asymptotics for the Riemann Theta Kernelanalytic number theory
- Exact-Gamma Four-Swap Asymptotics for Long Mellin-Radial Amplifiers in the Gaussian Angular Hecke Familyanalytic number theory
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“Third-level log-concavity of the Riemann Xi kernel.” TheoremDB. P11158. Problem statement; statement identity tdbc1:0881826e8eed9e05e1927a48ff7c039e54bad989cecdf5763ec6129b8c904222; statement text SHA-256 c34fe3e57ad2f3a7ccd1674d2b95abe371f92a33aec3dbed47ebb46fefb85e78. https://theoremdb.org/statement/?ref=P11158
@misc{theoremdb-problem-c34fe3e57ad2f3a7ccd1674d2b95abe371f92a33aec3dbed47ebb46fefb85e78,
title = {{Third-level log-concavity of the Riemann Xi kernel}},
howpublished = {TheoremDB},
note = {Problem statement; statement identity tdbc1:0881826e8eed9e05e1927a48ff7c039e54bad989cecdf5763ec6129b8c904222; statement text SHA-256 c34fe3e57ad2f3a7ccd1674d2b95abe371f92a33aec3dbed47ebb46fefb85e78},
url = {https://theoremdb.org/statement/?ref=P11158}
}Plain text: Built Markdown snapshot
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