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Sharp order-four sign regularity of a Jacobi theta kernel

A mix of human-written and AI-generated textHuman understanding: some partsmath.CA — Classical Analysis and ODEsmath.NT — Number Theory

Contributed by Suvrit Sra ↗

Version 1 / Nov 02, 2026 / CC BY 4.0

Abstract

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

Provenance statement

The results in this paper were developed with an AI assistant (OpenAI's GPT Pro). None of the claims were obtained zero-shot: they emerged through sustained back-and-forth discussion, including failed attempts, counterexamples, audits, and repairs. The compact verification in Section 3.3 is computer-assisted and has not been checked by hand by the author.

Tools used

OpenAI
GPT Pro

References

  1. Mark W. Coffey and George Csordas. On the log-concavity of a J acobi theta function. Mathematics of Computation , 82 0 (284): 0 2265--2272, 2013. 10.1090/S0025-5718-2013-02681-6.DOI
  2. George Csordas. Fourier transforms of positive definite kernels and the Riemann ξ\xi-function. Computational Methods and Function Theory , 15 0 (3): 0 373--391, 2015. 10.1007/s40315-014-0105-8.DOI
  3. George Csordas and Dimitar K. Dimitrov. Conjectures and theorems in the theory of entire functions. Numerical Algorithms , 25 0 (1--4): 0 109--122, 2000. 10.1023/A:1016604906346.DOI
  4. George Csordas and Richard S. Varga. Moment inequalities and the Riemann hypothesis. Constructive Approximation , 4 0 (1): 0 175--198, 1988. 10.1007/BF02075457.DOI
  5. George Csordas, Timothy S. Norfolk, and Richard S. Varga. The Riemann hypothesis and the Turán inequalities. Transactions of the American Mathematical Society , 296 0 (2): 0 521--541, 1986. 10.1090/S0002-9947-1986-0846596-4.DOI
  6. Samuel Karlin. Total Positivity, Vol. I . Stanford University Press, Stanford, CA, 1968.
  7. John Nuttall. Wronskians, cumulants, and the Riemann hypothesis. Constructive Approximation , 38 0 (2): 0 193--212, 2013. 10.1007/s00365-013-9200-8.DOI
  8. Michel Planat and Patrick Sol é . Second-level concavity of the R iemann Ξ\Xi kernel. arXiv:2608.19160 [math.CV], 2026. Posted 19 August 2026.arXiv

Version history

  1. v1Submitted by Suvrit SraInitial depositCurrentOct 31, 2026