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Let $\\Delta(x;\\ell_1,M_1,\\ell_2,M_2)$ be the error term of the asymptotic formula of $\\sum\\limits_{n\\leqslant x}d(n;\\ell_1,M_1,\\ell_2,M_2)$. In this paper, we establish an asymptotic formula of the fourth power moment of $\\Delta(M_1M_2x;\\ell_1,M_1,\\ell_2,M_2)$ and prove that \\begin{equation*}\n  \\int_1^T\\Delta^4(M_1M_2x;\\ell_1,M_1,\\ell_2,M_2)\\mathrm{d}x=\\frac{1}{32\\pi^4}C_4\\Big(\\frac{\\ell_1}{M_1},\\frac{"},"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":"1711.10808","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-11-29T12:19:00Z","cross_cats_sorted":[],"title_canon_sha256":"03f0c5091a7b3b787d4a3a7f30a49d537c93a497828a4c93e3128c87641fd146","abstract_canon_sha256":"3eef040c9db9d868b4fa6223fb3fc7301eb896f8d91c73c008f6a10625c07178"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:29:16.879215Z","signature_b64":"+N03HNy5WgbFIrrHcH8pCfbsGlKox5lf+ZXWlgwHwUg52iyCjoKc1GxpN4V5YB1GgDHBBOVA7UAQS68UMue8Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b44480c4c872fa0f15f96de7413ff2ac80bcaf47be7c603f854c0e70e640ce29","last_reissued_at":"2026-05-18T00:29:16.878715Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:29:16.878715Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Fourth Power Moment of the Error Term for the Divisor Problem with Congruence Conditions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jinjiang Li, Min Zhang","submitted_at":"2017-11-29T12:19:00Z","abstract_excerpt":"Let $d(n;\\ell_1,M_1,\\ell_2,M_2)$ denote the number of factorizations $n=n_1n_2$, where each of the factors $n_i\\in\\mathbb{N}$ belongs to a prescribed congruence class $\\ell_i\\bmod M_i\\,(i=1,2)$. Let $\\Delta(x;\\ell_1,M_1,\\ell_2,M_2)$ be the error term of the asymptotic formula of $\\sum\\limits_{n\\leqslant x}d(n;\\ell_1,M_1,\\ell_2,M_2)$. In this paper, we establish an asymptotic formula of the fourth power moment of $\\Delta(M_1M_2x;\\ell_1,M_1,\\ell_2,M_2)$ and prove that \\begin{equation*}\n  \\int_1^T\\Delta^4(M_1M_2x;\\ell_1,M_1,\\ell_2,M_2)\\mathrm{d}x=\\frac{1}{32\\pi^4}C_4\\Big(\\frac{\\ell_1}{M_1},\\frac{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.10808","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":"1711.10808","created_at":"2026-05-18T00:29:16.878792+00:00"},{"alias_kind":"arxiv_version","alias_value":"1711.10808v1","created_at":"2026-05-18T00:29:16.878792+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1711.10808","created_at":"2026-05-18T00:29:16.878792+00:00"},{"alias_kind":"pith_short_12","alias_value":"WRCIBRGIOL5A","created_at":"2026-05-18T12:31:53.515858+00:00"},{"alias_kind":"pith_short_16","alias_value":"WRCIBRGIOL5A6FPZ","created_at":"2026-05-18T12:31:53.515858+00:00"},{"alias_kind":"pith_short_8","alias_value":"WRCIBRGI","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/WRCIBRGIOL5A6FPZNXTUCP7SVS","json":"https://pith.science/pith/WRCIBRGIOL5A6FPZNXTUCP7SVS.json","graph_json":"https://pith.science/api/pith-number/WRCIBRGIOL5A6FPZNXTUCP7SVS/graph.json","events_json":"https://pith.science/api/pith-number/WRCIBRGIOL5A6FPZNXTUCP7SVS/events.json","paper":"https://pith.science/paper/WRCIBRGI"},"agent_actions":{"view_html":"https://pith.science/pith/WRCIBRGIOL5A6FPZNXTUCP7SVS","download_json":"https://pith.science/pith/WRCIBRGIOL5A6FPZNXTUCP7SVS.json","view_paper":"https://pith.science/paper/WRCIBRGI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1711.10808&json=true","fetch_graph":"https://pith.science/api/pith-number/WRCIBRGIOL5A6FPZNXTUCP7SVS/graph.json","fetch_events":"https://pith.science/api/pith-number/WRCIBRGIOL5A6FPZNXTUCP7SVS/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WRCIBRGIOL5A6FPZNXTUCP7SVS/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WRCIBRGIOL5A6FPZNXTUCP7SVS/action/storage_attestation","attest_author":"https://pith.science/pith/WRCIBRGIOL5A6FPZNXTUCP7SVS/action/author_attestation","sign_citation":"https://pith.science/pith/WRCIBRGIOL5A6FPZNXTUCP7SVS/action/citation_signature","submit_replication":"https://pith.science/pith/WRCIBRGIOL5A6FPZNXTUCP7SVS/action/replication_record"}},"created_at":"2026-05-18T00:29:16.878792+00:00","updated_at":"2026-05-18T00:29:16.878792+00:00"}