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We obtain $$\\Delta(T)=\\Omega\\left(T^{\\alpha(T)}\\right) \\text{ where } \\alpha(T) =\\frac{3}{8}-\\frac{c}{(\\log T)^{1/8}} \\text{ and } c>0,$$ along with an $\\Omega$-bound for th"},"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":"1807.10047","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-07-26T10:06:52Z","cross_cats_sorted":[],"title_canon_sha256":"4d435bf836aac4d3469cb36182529a73f6891ba7613c086f82104023971ef0dc","abstract_canon_sha256":"15e14cb970914304dc1ec115dcaf9db2cf53d0a6dce613478fb0cd98ea44dbfd"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:09:45.565104Z","signature_b64":"zcslIjchdpbHaOKo20HhM9NzH8HChPVJnYAq1Z5GkNbDOrwIqQt3zxWbqUhnIdMSms+UTYB/zJQKAuuCmSB+CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4af6ffc19912281e4596b99dcf63df79e4f85da9aeb4c659d2d6466935bddcf6","last_reissued_at":"2026-05-18T00:09:45.564384Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:09:45.564384Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Omega Theorems for The Twisted Divisor Function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Anirban Mukhopadhyay, Kamalakshya Mahatab","submitted_at":"2018-07-26T10:06:52Z","abstract_excerpt":"For a fixed $\\theta\\neq 0$, we define the twisted divisor function $$ \\tau(n, \\theta):=\\sum_{d\\mid n}d^{i\\theta}\\ .$$ In this article we consider the error term $\\Delta(x)$ in the following asymptotic formula $$ \\sum_{n\\leq x}^*|\\tau(n, \\theta)|^2=\\omega_1(\\theta)x\\log x + \\omega_2(\\theta)x\\cos(\\theta\\log x) +\\omega_3(\\theta)x + \\Delta(x),$$ where $\\omega_i(\\theta)$ for $i=1, 2, 3$ are constants depending only on $\\theta$. We obtain $$\\Delta(T)=\\Omega\\left(T^{\\alpha(T)}\\right) \\text{ where } \\alpha(T) =\\frac{3}{8}-\\frac{c}{(\\log T)^{1/8}} \\text{ and } c>0,$$ along with an $\\Omega$-bound for th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.10047","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":""},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1807.10047","created_at":"2026-05-18T00:09:45.564491+00:00"},{"alias_kind":"arxiv_version","alias_value":"1807.10047v1","created_at":"2026-05-18T00:09:45.564491+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1807.10047","created_at":"2026-05-18T00:09:45.564491+00:00"},{"alias_kind":"pith_short_12","alias_value":"JL3P7QMZCIUB","created_at":"2026-05-18T12:32:31.084164+00:00"},{"alias_kind":"pith_short_16","alias_value":"JL3P7QMZCIUB4RMW","created_at":"2026-05-18T12:32:31.084164+00:00"},{"alias_kind":"pith_short_8","alias_value":"JL3P7QMZ","created_at":"2026-05-18T12:32:31.084164+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/JL3P7QMZCIUB4RMWXGO46Y67PH","json":"https://pith.science/pith/JL3P7QMZCIUB4RMWXGO46Y67PH.json","graph_json":"https://pith.science/api/pith-number/JL3P7QMZCIUB4RMWXGO46Y67PH/graph.json","events_json":"https://pith.science/api/pith-number/JL3P7QMZCIUB4RMWXGO46Y67PH/events.json","paper":"https://pith.science/paper/JL3P7QMZ"},"agent_actions":{"view_html":"https://pith.science/pith/JL3P7QMZCIUB4RMWXGO46Y67PH","download_json":"https://pith.science/pith/JL3P7QMZCIUB4RMWXGO46Y67PH.json","view_paper":"https://pith.science/paper/JL3P7QMZ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1807.10047&json=true","fetch_graph":"https://pith.science/api/pith-number/JL3P7QMZCIUB4RMWXGO46Y67PH/graph.json","fetch_events":"https://pith.science/api/pith-number/JL3P7QMZCIUB4RMWXGO46Y67PH/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/JL3P7QMZCIUB4RMWXGO46Y67PH/action/timestamp_anchor","attest_storage":"https://pith.science/pith/JL3P7QMZCIUB4RMWXGO46Y67PH/action/storage_attestation","attest_author":"https://pith.science/pith/JL3P7QMZCIUB4RMWXGO46Y67PH/action/author_attestation","sign_citation":"https://pith.science/pith/JL3P7QMZCIUB4RMWXGO46Y67PH/action/citation_signature","submit_replication":"https://pith.science/pith/JL3P7QMZCIUB4RMWXGO46Y67PH/action/replication_record"}},"created_at":"2026-05-18T00:09:45.564491+00:00","updated_at":"2026-05-18T00:09:45.564491+00:00"}