{"paper":{"title":"On Shusterman's Goldbach-type problem for sign patterns of the Liouville function","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Alexander P. Mangerel","submitted_at":"2024-12-23T00:26:52Z","abstract_excerpt":"Let $\\lambda$ be the Liouville function. Assuming the Generalised Riemann Hypothesis for Dirichlet $L$-functions (GRH), we show that for every sufficiently large even integer $N$ there are $a,b \\geq 1$ such that $$ a+b = N \\text{ and } \\lambda(a) = \\lambda(b) = -1. $$ This conditionally answers an analogue of the binary Goldbach problem for the Liouville function, posed by Shusterman.\n  The latter is a consequence of a quantitative lower bound on the frequency of sign patterns attained by $(\\lambda(n),\\lambda(N-n))$, for sufficiently large primes $N$. We show, assuming GRH, that there is a con"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.17199","kind":"arxiv","version":1},"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/2412.17199/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"}