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A counterexample to Haglund's monotonic-zeros conjecture for the Riemann Ξ\Xi approximants

Primarily AI-generated textHuman understanding: some partsmath.NT — Number Theory

Contributed by Kunal Tyagi ↗

Version 1 / Oct 14, 2026 / CC BY 4.0

Abstract

Haglund introduced the partial sums ΞN=∑n≤NΦn\Xi_N = \sum_{n \le N} \Phi_n of Riemann's expansion of the Ξ\Xi-function in incomplete gamma functions and conjectured that for every NN the zeros of ΞN\Xi_N in the closed first quadrant, listed by increasing real part, have nondecreasing imaginary parts. The conjecture implies the Riemann hypothesis; Haglund's computations supported a weak form of it for N≤10N \le 10. We show that the conjecture is false for N=27N = 27: Ξ27\Xi_{27} has a real zero in (3144.8946,3144.8947)(3144.8946, 3144.8947) and a non-real zero z0=3143.2206824215…+0.3152587993… iz_0 = 3143.2206824215\ldots + 0.3152587993\ldots\, i in the first quadrant, whose real part is smaller. The weak form fails as well. Both facts are certified twice, independently, in interval arithmetic (Arb ball arithmetic; outward-rounded intervals with proved truncation bounds), and z0z_0 is enclosed in a square of half-width 10−4810^{-48}. The same violation occurs at N=24N = 24 for the companion approximants ξN=ΞN+cN\xi_N = \Xi_N + c_N. The failure has a mechanism: on the critical line ΞN<Ξ<ξN\Xi_N < \Xi < \xi_N, so the real zeros of ΞN\Xi_N lie in positive lobes of Ξ\Xi that rise above a positive level, and a low lobe followed by a higher one near the height 4(N+1)24(N+1)^2, where the approximant departs from Ξ\Xi, can leave a non-real zero below real zeros. The same structure shows that the number of positive real zeros of ΞN\Xi_N, counted with multiplicity, is odd, and that x2ΞN(x)x^2\Xi_N(x) tends to a negative limit on the real axis for every NN.

Provenance statement

The mathematics in this paper - the derivations, the computations, the verification suite and the text - was produced by Claude (Anthropic) working under the author's direction inside a multi-session research program; the author set the objectives, made the technical and editorial decisions, and is responsible for the content. The counterexample is certified by two independent interval-arithmetic computations (Arb ball arithmetic through python-flint; outward-rounded intervals with proved truncation bounds), written separately, and the final text was checked against its verified sources by a second automated pass before posting. The complete working record (code, logs, certificates, session journal) is public at https://github.com/x67ai/riemann-rh-program. The same version of this paper is deposited on Zenodo, DOI 10.5281/zenodo.23071930 (version 2).

Tools used

Anthropic
Claude FableVersion 5.1 (orchestration), October 2026
Anthropic
Claude OpusVersion 5.5 (agents), October 2026
Anthropic
Claude CodeVersion CLI, October 2026
FLINT project
Arb ball arithmetic via python-flintVersion python-flint 0.6.0

References

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Version history

  1. v1Submitted by Kunal TyagiInitial depositCurrentOct 14, 2026