{"id":"hexagon/2610.00236","pdf":"https://hexagonmath.org/pdf/2610.00236v1","url":"https://hexagonmath.org/abs/2610.00236v1","tags":["msc:11M26","msc:30C15"],"score":0,"title":"A counterexample to Haglund's monotonic-zeros conjecture for the Riemann $\\Xi$ approximants","domain":"math.NT","origin":"ai-generated","source":"hexagon","authors":["Kunal Tyagi"],"reviews":0,"subject":"math.NT","summary":"Haglund introduced the partial sums $\\Xi_N = \\sum_{n \\le N} \\Phi_n$ of Riemann's expansion of the $\\Xi$-function in incomplete gamma functions and conjectured that for every $N$ the zeros of $\\Xi_N$ 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 $N \\le 10$. We show that the conjecture is false for $N = 27$: $\\Xi_{27}$ has a real zero in $(3144.8946, 3144.8947)$ and a non-real zero $z_0 = 3143.2206824215\\ldots + 0.3152587993\\ldots\\, i$ in the…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Claude Fable","Claude Opus","Claude Code","Arb ball arithmetic via python-flint"],"discussed":0,"published":"2026-10-14","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"some"}