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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."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2406.12283","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2024-06-18T05:22:10Z","cross_cats_sorted":[],"title_canon_sha256":"aaacea7f77130f61ca0c3d3c13f2846560b87217ea886056a4db9a358ea2404c","abstract_canon_sha256":"b3f6f71a012be8fa5e9f4bda5023bc114d3a70be960520a42cf15b080be861b0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-09T01:05:03.453981Z","signature_b64":"tgGFRouw3B//2pN9/v5ut4nyTua4M9ZB/kP/VODmFsiRim6tB6JAQSDg6Yl/Ge6J+Qv12QcLnu+SxXTMcBtYDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0981082b526912042092738cc6d43d533e933911324cd499e8634c59112d6270","last_reissued_at":"2026-06-09T01:05:03.453493Z","signature_status":"signed_v1","first_computed_at":"2026-06-09T01:05:03.453493Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $a$. 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