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These bounds are sharper than the ones which follow by the Cauchy-Schwarz inequality and mean square results for $\\Delta_k(x)$. We also obtain the analogues of the above bounds when $\\D(x)$ is replaced by $E(x)$, the error term in the mean square formula for $|\\zeta(1/2+it)|$."},"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":"1711.09589","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-11-27T09:15:39Z","cross_cats_sorted":[],"title_canon_sha256":"7d03b90e4887a3fcdf855919461678c603a9e7d15c08d7d4680add2f4f9ed557","abstract_canon_sha256":"9dbe5771b91b3936529b46a93e76b8059465edcd01db272683a055db9f73f8e0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:29:35.065839Z","signature_b64":"D9rZF634sdIedwPnMClzDUGZT+rixGhTQw2NI5wjfLa6W+qC+8sX1aKifZI9+1YPZjvv/NR8IOJp4fLDNHYZBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b2a7b5c59e0cd9b89250c7f34818c561805eb81140c7686092895b62d901c94c","last_reissued_at":"2026-05-18T00:29:35.065153Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:29:35.065153Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On certain integrals involving the Dirichlet divisor problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Aleksandar Ivi\\'c, Wenguang Zhai","submitted_at":"2017-11-27T09:15:39Z","abstract_excerpt":"We prove that $$ \\int_1^X\\Delta(x)\\Delta_3(x)\\,dx \\ll X^{13/9}\\log^{10/3}X, \\quad \\int_1^X\\Delta(x)\\Delta_4(x)\\,dx \\ll_\\varepsilon X^{25/16+\\varepsilon}, $$ where $\\Delta_k(x)$ is the error term in the asymptotic formula for the summatory function of $d_k(n)$, generated by $\\zeta^k(s)$ ($\\Delta_2(x) \\equiv \\Delta(x)$). These bounds are sharper than the ones which follow by the Cauchy-Schwarz inequality and mean square results for $\\Delta_k(x)$. We also obtain the analogues of the above bounds when $\\D(x)$ is replaced by $E(x)$, the error term in the mean square formula for $|\\zeta(1/2+it)|$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.09589","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":"1711.09589","created_at":"2026-05-18T00:29:35.065252+00:00"},{"alias_kind":"arxiv_version","alias_value":"1711.09589v1","created_at":"2026-05-18T00:29:35.065252+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1711.09589","created_at":"2026-05-18T00:29:35.065252+00:00"},{"alias_kind":"pith_short_12","alias_value":"WKT3LRM6BTM3","created_at":"2026-05-18T12:31:53.515858+00:00"},{"alias_kind":"pith_short_16","alias_value":"WKT3LRM6BTM3RESQ","created_at":"2026-05-18T12:31:53.515858+00:00"},{"alias_kind":"pith_short_8","alias_value":"WKT3LRM6","created_at":"2026-05-18T12:31:53.515858+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/WKT3LRM6BTM3RESQY7ZUQGGFMG","json":"https://pith.science/pith/WKT3LRM6BTM3RESQY7ZUQGGFMG.json","graph_json":"https://pith.science/api/pith-number/WKT3LRM6BTM3RESQY7ZUQGGFMG/graph.json","events_json":"https://pith.science/api/pith-number/WKT3LRM6BTM3RESQY7ZUQGGFMG/events.json","paper":"https://pith.science/paper/WKT3LRM6"},"agent_actions":{"view_html":"https://pith.science/pith/WKT3LRM6BTM3RESQY7ZUQGGFMG","download_json":"https://pith.science/pith/WKT3LRM6BTM3RESQY7ZUQGGFMG.json","view_paper":"https://pith.science/paper/WKT3LRM6","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1711.09589&json=true","fetch_graph":"https://pith.science/api/pith-number/WKT3LRM6BTM3RESQY7ZUQGGFMG/graph.json","fetch_events":"https://pith.science/api/pith-number/WKT3LRM6BTM3RESQY7ZUQGGFMG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WKT3LRM6BTM3RESQY7ZUQGGFMG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WKT3LRM6BTM3RESQY7ZUQGGFMG/action/storage_attestation","attest_author":"https://pith.science/pith/WKT3LRM6BTM3RESQY7ZUQGGFMG/action/author_attestation","sign_citation":"https://pith.science/pith/WKT3LRM6BTM3RESQY7ZUQGGFMG/action/citation_signature","submit_replication":"https://pith.science/pith/WKT3LRM6BTM3RESQY7ZUQGGFMG/action/replication_record"}},"created_at":"2026-05-18T00:29:35.065252+00:00","updated_at":"2026-05-18T00:29:35.065252+00:00"}