Cubic sieves, a primitive-ratio cut and a 20/37 bound for the abc exceptional set
Ruiyi Zhang · Preprint, 2026-10-09 · 41 pages
PDFHTMLDOI 10.5281/zenodo.23266656Source and checks
Abstract
We prove N_1(X) ≪_ε X^(20/37+ε), where N_λ(X) counts the abc triples of exponent λ in [1, X]^3. The previous full bound was X^(13/20+ε). We use the anatomic reduction of Bernert, Browning, Lichtman and Teräväinen. Five families add new rows. The first two are an archimedean cover and a row from Heath-Brown's uniform bound for diagonal cubic curves. Spacing gives 30 two-arm clauses. Cubic characters and a fourth moment of Dirichlet L-functions give the cubic-sieve rows. Cochrane and Shi's fourth moment gives the moment rows. An exact Farkas certificate with 394 633 leaves gives exponent 20/37 for the box count. On each leaf the multipliers of all rows except nonnegativity sum to at most 586/201. Exact rows are included in this sum. This pays for the losses at λ = 1.
Cite
Ruiyi Zhang, Cubic sieves, a primitive-ratio cut and a 20/37 bound for the abc exceptional set, preprint, Zenodo, 2026. https://doi.org/10.5281/zenodo.23266656