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It is shown that $$ \\int_0^T\\Delta(t)|\\zeta(1/2+it)|^2\\,dt \\ll T(\\log T)^{4}. $$ Further, if $2\\le k\\le 8$ is a fixed integer, then we prove the asymptotic formula $$ \\int_1^{T}\\Delta^{k}(t)|\\zeta(1/2+it)|^2\\,dt=c_1(k)T^{1+\\frac k4}\\log T+ c_2(k)T^{1+\\frac k4}+O_\\varepsilon(T^{1+\\frac k4-\\eta_k+\\varepsilon}), $$ where $c_1(k)$ and "},"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":"1502.00406","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-02-02T09:04:55Z","cross_cats_sorted":[],"title_canon_sha256":"e28350b8004213cd528652441d970bbd6fc67eb8159f5fa420438e0a58673699","abstract_canon_sha256":"32db27873a5e5f02e50f4fb8b3fa76561815547622324a14f1c9139ce38d3aea"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:25:18.318403Z","signature_b64":"X26QWTxqR0ZrvcKpP4BvU+QvwZeat3irsJOuK+5meX/g97FRaa1zz/Lvjgf4hKjUkC9p8c0EBpS+0Xdwy7qDDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"82fb0882c490739c7824cf2a85499003cd368a4d443533762136167ae079c5b1","last_reissued_at":"2026-05-18T01:25:18.317681Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:25:18.317681Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On some mean value results for the zeta-function and a divisor problem II","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":"2015-02-02T09:04:55Z","abstract_excerpt":"Let $d(n)$ be the number of divisors of $n$, let $\\gamma$ denote Euler's constant and $$ \\Delta(x) := \\sum_{n\\le x}d(n) - x(\\log x + 2\\gamma -1) $$ denote the error term in the classical Dirichlet divisor problem, and let $\\zeta(s)$ denote the Riemann zeta-function. It is shown that $$ \\int_0^T\\Delta(t)|\\zeta(1/2+it)|^2\\,dt \\ll T(\\log T)^{4}. $$ Further, if $2\\le k\\le 8$ is a fixed integer, then we prove the asymptotic formula $$ \\int_1^{T}\\Delta^{k}(t)|\\zeta(1/2+it)|^2\\,dt=c_1(k)T^{1+\\frac k4}\\log T+ c_2(k)T^{1+\\frac k4}+O_\\varepsilon(T^{1+\\frac k4-\\eta_k+\\varepsilon}), $$ where $c_1(k)$ and "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.00406","kind":"arxiv","version":2},"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":"1502.00406","created_at":"2026-05-18T01:25:18.317819+00:00"},{"alias_kind":"arxiv_version","alias_value":"1502.00406v2","created_at":"2026-05-18T01:25:18.317819+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1502.00406","created_at":"2026-05-18T01:25:18.317819+00:00"},{"alias_kind":"pith_short_12","alias_value":"QL5QRAWESBZZ","created_at":"2026-05-18T12:29:37.295048+00:00"},{"alias_kind":"pith_short_16","alias_value":"QL5QRAWESBZZY6BE","created_at":"2026-05-18T12:29:37.295048+00:00"},{"alias_kind":"pith_short_8","alias_value":"QL5QRAWE","created_at":"2026-05-18T12:29:37.295048+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/QL5QRAWESBZZY6BEZ4VIKSMQAP","json":"https://pith.science/pith/QL5QRAWESBZZY6BEZ4VIKSMQAP.json","graph_json":"https://pith.science/api/pith-number/QL5QRAWESBZZY6BEZ4VIKSMQAP/graph.json","events_json":"https://pith.science/api/pith-number/QL5QRAWESBZZY6BEZ4VIKSMQAP/events.json","paper":"https://pith.science/paper/QL5QRAWE"},"agent_actions":{"view_html":"https://pith.science/pith/QL5QRAWESBZZY6BEZ4VIKSMQAP","download_json":"https://pith.science/pith/QL5QRAWESBZZY6BEZ4VIKSMQAP.json","view_paper":"https://pith.science/paper/QL5QRAWE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1502.00406&json=true","fetch_graph":"https://pith.science/api/pith-number/QL5QRAWESBZZY6BEZ4VIKSMQAP/graph.json","fetch_events":"https://pith.science/api/pith-number/QL5QRAWESBZZY6BEZ4VIKSMQAP/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/QL5QRAWESBZZY6BEZ4VIKSMQAP/action/timestamp_anchor","attest_storage":"https://pith.science/pith/QL5QRAWESBZZY6BEZ4VIKSMQAP/action/storage_attestation","attest_author":"https://pith.science/pith/QL5QRAWESBZZY6BEZ4VIKSMQAP/action/author_attestation","sign_citation":"https://pith.science/pith/QL5QRAWESBZZY6BEZ4VIKSMQAP/action/citation_signature","submit_replication":"https://pith.science/pith/QL5QRAWESBZZY6BEZ4VIKSMQAP/action/replication_record"}},"created_at":"2026-05-18T01:25:18.317819+00:00","updated_at":"2026-05-18T01:25:18.317819+00:00"}