{"paper":{"title":"Note on a sum involving the divisor function","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Liuying Wu","submitted_at":"2025-05-03T01:19:00Z","abstract_excerpt":"Let $d(n)$ be the divisor function and denote by $[t]$ the integral part of the real number $t$. In this paper, we prove that $$\\sum_{n\\leq x^{1/c}}d\\left(\\left[\\frac{x}{n^c}\\right]\\right)=d_cx^{1/c}+\\mathcal{O}_{\\varepsilon,c} \\left(x^{\\max\\{(2c+2)/(2c^2+5c+2),5/(5c+6)\\}+\\varepsilon}\\right),$$ where $d_c=\\sum_{k\\geq1}d(k)\\left(\\frac{1}{k^{1/c}}-\\frac{1}{(k+1)^{1/c}}\\right)$ is a constant. This result constitutes an improvement upon that of Feng."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.01645","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/2505.01645/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"}