Han–Xiong 1/2-conjecture

The Han–Xiong conjecture asserts that the integer trace at $r=\frac12$ coefficientwise dominates the corresponding trace for every positive rational $r$ and integer $k\ge1$. The original paper gives special cases and computational evidence, not a complete proof.

Known results

• Han and Xiong: proved several special families, including integral and half-integral $r$, and established the conjecture for all positive rational $r$ when $k\le3$.
• Exact computation verifies all positive rational $r$ for $1\le k\le200$; computations for $r=\frac13$ and $r=\frac14$ reach $k=150$.

September 2026 developments

A new manuscript proves a support-dominance theorem implying the conjecture for every rational $r\ge\frac12$, reduces the remaining problem to primitive parameters $r=\frac1{2m}$, and identifies $r=\frac14$ as the first unresolved primitive case. Its all-$k$ conclusions are reported as formally verified by AxiomProver, but the full conjecture remains unproved. On September 8, 2026, George E. Andrews and Mohamed El Bachraoui published adjacent-pair estimates in The Ramanujan Journal; the supplied record gives no abstract, so their precise strength cannot be assessed.

Current status (as of September 2026): substantial special cases, a reduction to $r=\frac1{2m}$, and finite computations are available, while the all-$k$ conjecture remains open and the latest estimates do not indicate a complete resolution.