Research Preprints · Real and complex analysis

Sharp order-four sign regularity of a Jacobi theta kernel

Suvrit Sra · TU Munich, Department of Mathematics, School of CIT

Version v1 · released 10 October 2026 · 11 pages · Reverse regularity of the square-root theta kernel and Csordas's conjecture

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Abstract

We prove that the Hankel kernel $\Phi(\sqrt{x+y})$, where $\Phi$ is the Jacobi theta kernel whose cosine transform is the Riemann $\Xi$-function, is strictly reverse-regular through order four. The order is sharp: an interval enclosure of the fifth Hankel–Wronskian at $t=10^{-2}$ shows failure at order five. The third-order condition is Csordas's conjecture that $s'^2-ss''$ is strictly log-concave for $s(t)=\Phi(\sqrt t)$. The proof reduces the kernel minors to Wronskian signs and establishes the signs by exact polynomial estimates for $t\ge1$ and interval arithmetic on $[0,1]$. The fourth-order sign also yields three positive iterates of $s\mapsto s'^2-ss''$, the triple Turán inequalities for the normalized Taylor coefficients of $\Xi$, and quartic integral inequalities for $\Xi$.

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@misc{sra2026csordas-v1,
  author       = {Suvrit Sra},
  title        = {{Sharp order-four sign regularity of a Jacobi theta kernel}},
  year         = {2026},
  month        = oct,
  howpublished = {Preprint, \url{https://github.com/suvrit/preprints/tree/main/preprints/csordas/v1}},
  note         = {Version v1, released 10 October 2026},
}

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