{"paper":{"title":"Hidden Multiscale Order in the Primes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Ge Zhang, Matthew de Courcy-Ireland, Salvatore Torquato","submitted_at":"2018-04-16T00:49:02Z","abstract_excerpt":"We study the {pair correlations between} prime numbers in an interval $M \\leq p \\leq M + L$ with $M \\rightarrow \\infty$, $L/M \\rightarrow \\beta > 0$. By analyzing the \\emph{structure factor}, we prove, conditionally on the {Hardy-Littlewood conjecture on prime pairs}, that the primes are characterized by unanticipated multiscale order. Specifically, their limiting structure factor is that of a union of an infinite number of periodic systems and is characterized by dense set of Dirac delta functions. Primes in dyadic intervals are the first examples of what we call {\\it effectively limit-period"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.06279","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":""},"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"}