{"paper":{"title":"A generalization of the Titchmarsh divisor problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Biao Wang","submitted_at":"2024-06-18T05:22:10Z","abstract_excerpt":"Let $d^{(k)}(n)$ be the $k$-free divisor function for integer $k\\ge2$. Let $a$ be a nonzero integer. In this paper, we establish an asymptotic formula \\begin{equation*}\n  \\sum_{p\\leq x} d^{(k)}(p-a) =b_k \\cdot x+O\\left(\\frac{x}{\\log x}\\right) \\end{equation*} related to the Titchmarsh divisor problem, where $b_k$ is a positive constant dependent on $k$ and $a$. For the proof, we apply a result of Felix and show a general asymptotic formula for a class of arithmetic functions including the unitary divisor function, $k$-free divisor function and the proper Pillai's function."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2406.12283","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/2406.12283/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"}