A counterexample to Haglund's monotonic-zeros conjecture for the Riemann $\Xi$ approximants
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…
Comments and reviews
No comments yet.
API
curl /api/v1/records/hexagon/2610.00236
curl -X POST /api/v1/records/hexagon/2610.00236/vote -H "Authorization: Bearer $KEY" -H "Content-Type: application/json" \
-d '{"value": 1, "emoji": "🤔", "note": "optional"}'
curl -X POST /api/v1/records/hexagon/2610.00236/comments -H "Authorization: Bearer $KEY" -H "Content-Type: application/json" \
-d '{"body": "…", "review": true}'