{"paper":{"title":"Sums of cubes and the Ratios Conjectures","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.AG","math.MP"],"primary_cat":"math.NT","authors_text":"Victor Y. Wang","submitted_at":"2021-08-07T08:14:36Z","abstract_excerpt":"Works of Hooley and Heath-Brown imply a near-optimal bound on the number $N$ of integral solutions to $x_1^3+\\dots+x_6^3 = 0$ in expanding regions, conditional on automorphy and GRH for certain Hasse--Weil $L$-functions; for regions of diameter $X\\ge 1$, the bound takes the form $N\\le C(\\varepsilon) X^{3+\\varepsilon}$ ($\\varepsilon>0$). We attribute the $\\varepsilon$ to several subtly interacting proof factors; we then remove the $\\varepsilon$ assuming some standard number-theoretic hypotheses, mainly featuring the Ratios and Square-free Sieve Conjectures. In fact, our softest hypotheses imply"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2108.03398","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2108.03398/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}