Sub-study of hunt #38 (hunts/rogue_frontier/MISSION.md carries the governing HuntSpec). Opened 2026-08-17.
Question
Compute the Nyman-Beurling / Baez-Duarte distance
d_N^2 = inf_{a_1..a_N} || chi_(0,1) - sum_{k<=N} a_k {1/(k t)} ||^2_{L^2(0,infty)}
with rigorous ball arithmetic end to end: exact Gram entries by the Vasyunin cotangent formula, an arb ball solve of the normal equations, and an enclosure for every reported d_N^2. Push N as far as one session's budget allows (powers of 2), then measure the approach of d_N^2 * log N to the conjectured constant 2 + gamma - log(4 pi) = 0.0461914179... and the conditioning of the Gram matrix as a function of N.
Why it is worth a session
Literature numerics (Landreau-Richard 2002) are float runs with no error control. Prior exact work stops at the formula; nobody appears to publish d_N^2 with proven enclosures. The repo's own zeta/criteria.py computes a related but different quadratic form (the BBLS basis on L^2(0,1), k >= 2); this study is the L^2(0,infty) form the Bettin-Conrey-Farmer asymptotic actually addresses, and the two sequences cross-illuminate.
Method and oracles
- Gram entries: Vasyunin's formula (two independent published statements pinned in
SOURCES.md), implemented in python-flint arb balls. - Independent checks: mpmath reimplementation at 50+ digits, direct numerical integration of selected entries with a trigamma-exact tail, closed forms for the diagonal and the target inner products.
- Solve:
arb_mat.solve, which proves invertibility and returns enclosures. Cross-checked at two precisions, against an mpmath solve at small N, and against the residual quadratic form (a rigorous upper bound at any vector).
Kill conditions (inherited from the parent HuntSpec, specialized)
- Vasyunin values disagree with direct quadrature beyond enclosure widths.
- The arb solve and the mpmath solve disagree at small N.
- d_N^2 fails monotone decrease in N, or the residual form falls below the reported ball: either would mean a defect, and the run stops there.
Scope
Writes only inside hunts/rogue_frontier/nyman_beurling/. Grade of every number here: measured, with enclosure-checked arithmetic where balls are reported. Nothing in this directory is a result until it leaves hunts/ by a sanctioned route, and nothing here is evidence about the truth of RH (docs/08; Littlewood's caution applies with full force).