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For the reciprocal square sums \\begin{equation*}\n  S(n)=\\sum_{\\substack{r=1 \\\\ (r,n)=1}}^{\\lfloor n/e \\rfloor}\\frac{1}{r^2} \\end{equation*} we already know the form of the congruence modulo $n$. In this paper, motivated by the known congruences, we first extend these results to certain reciprocal sums of odd order and establish a uniform congruence modulo $n$ for \\begin{equation*}\n  S_m(n)=\\sum_{\\substack{r=1 \\\\ (r,n)=1}}^{\\lfloor n/e \\rfloor}\\fr"},"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":"2607.11113","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-07-13T05:43:03Z","cross_cats_sorted":[],"title_canon_sha256":"9643c3098841dd1eef5c951ebd1502e333bf67e32b1cb0847f8409055222a06b","abstract_canon_sha256":"ec94f28c7d0aada908106bf9d534afff4e398a3ad1acc55a4cc507055580ff84"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-14T01:22:17.109111Z","signature_b64":"lIdsTe0nVf/NcZW6zxlNwzVeUjXgdUod0wf9m33CJmD0cYde/2D+fyIddMP/Z0la3GdGCRiWqUL/jP8qkbXkBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4cef5abf5087ce0f052e4022854675b94d50d77047fcdb6d6a7b17acd19dddaa","last_reissued_at":"2026-07-14T01:22:17.108236Z","signature_status":"signed_v1","first_computed_at":"2026-07-14T01:22:17.108236Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Higher-Order Congruence for Reciprocal Power Sums and Generalized Lehmer-Type Products","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Hao Zhong, Zhenming Tang","submitted_at":"2026-07-13T05:43:03Z","abstract_excerpt":"This paper investigates high-order congruences of reciprocal power sums and Lehmer-type products. Let $n\\geq 1$ with $(n,6)=1$ and $e\\in\\{2,3,4,6\\}$. For the reciprocal square sums \\begin{equation*}\n  S(n)=\\sum_{\\substack{r=1 \\\\ (r,n)=1}}^{\\lfloor n/e \\rfloor}\\frac{1}{r^2} \\end{equation*} we already know the form of the congruence modulo $n$. In this paper, motivated by the known congruences, we first extend these results to certain reciprocal sums of odd order and establish a uniform congruence modulo $n$ for \\begin{equation*}\n  S_m(n)=\\sum_{\\substack{r=1 \\\\ (r,n)=1}}^{\\lfloor n/e \\rfloor}\\fr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.11113","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.11113/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2607.11113","created_at":"2026-07-14T01:22:17.108703+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.11113v1","created_at":"2026-07-14T01:22:17.108703+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.11113","created_at":"2026-07-14T01:22:17.108703+00:00"},{"alias_kind":"pith_short_12","alias_value":"JTXVVP2QQ7HA","created_at":"2026-07-14T01:22:17.108703+00:00"},{"alias_kind":"pith_short_16","alias_value":"JTXVVP2QQ7HA6BJO","created_at":"2026-07-14T01:22:17.108703+00:00"},{"alias_kind":"pith_short_8","alias_value":"JTXVVP2Q","created_at":"2026-07-14T01:22:17.108703+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/JTXVVP2QQ7HA6BJOIARIKRTVXF","json":"https://pith.science/pith/JTXVVP2QQ7HA6BJOIARIKRTVXF.json","graph_json":"https://pith.science/api/pith-number/JTXVVP2QQ7HA6BJOIARIKRTVXF/graph.json","events_json":"https://pith.science/api/pith-number/JTXVVP2QQ7HA6BJOIARIKRTVXF/events.json","paper":"https://pith.science/paper/JTXVVP2Q"},"agent_actions":{"view_html":"https://pith.science/pith/JTXVVP2QQ7HA6BJOIARIKRTVXF","download_json":"https://pith.science/pith/JTXVVP2QQ7HA6BJOIARIKRTVXF.json","view_paper":"https://pith.science/paper/JTXVVP2Q","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.11113&json=true","fetch_graph":"https://pith.science/api/pith-number/JTXVVP2QQ7HA6BJOIARIKRTVXF/graph.json","fetch_events":"https://pith.science/api/pith-number/JTXVVP2QQ7HA6BJOIARIKRTVXF/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/JTXVVP2QQ7HA6BJOIARIKRTVXF/action/timestamp_anchor","attest_storage":"https://pith.science/pith/JTXVVP2QQ7HA6BJOIARIKRTVXF/action/storage_attestation","attest_author":"https://pith.science/pith/JTXVVP2QQ7HA6BJOIARIKRTVXF/action/author_attestation","sign_citation":"https://pith.science/pith/JTXVVP2QQ7HA6BJOIARIKRTVXF/action/citation_signature","submit_replication":"https://pith.science/pith/JTXVVP2QQ7HA6BJOIARIKRTVXF/action/replication_record"}},"created_at":"2026-07-14T01:22:17.108703+00:00","updated_at":"2026-07-14T01:22:17.108703+00:00"}