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In this paper we proved the following Theorem A: Let $G$ be a non-cyclic group of odd order $n=qm$, where $q$ is the smallest prime divisor of $n$ and $(m,q)=1$. Then the following statements hold. (1) If $q=3$, then $\\frac {\\psi(G)}{\\psi(|G|)}\\leq \\frac {85}{301}$, and equality holds if and only if $n=3\\cdot 7\\cdot m_1$ with $(m_1,42)=1$ and $G=(C_7\\rtimes C_3)\\times C_{m_1}$, with $C_7\\rtimes C_3$ non-abelia"},"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":"1905.12291","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2019-05-29T09:42:30Z","cross_cats_sorted":[],"title_canon_sha256":"2e75faba8a4e943d2c690c17dcfe1502d709369ab0c1425197674aab1ff6a7ba","abstract_canon_sha256":"b55f70b9e54fd9440b475662bca20c9dd3a4de56df192e0ab64144c377fcfe2a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:44:44.734703Z","signature_b64":"6QpmKbSu5Mox9/cwvqABLyrkSSY5xlrZDZiHmaEQLBsbJT48pTuvSGbz/u0kWMlzVgUYj5k2EMYxkkUKFdZJCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a407f533b62ee6fbc35b8a234e199c770d19f31a303425715b831130f5249010","last_reissued_at":"2026-05-17T23:44:44.733979Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:44:44.733979Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Sums of element orders in groups of odd order","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Marcel Herzog, Mercede Maj, Patrizia Longobardi","submitted_at":"2019-05-29T09:42:30Z","abstract_excerpt":"Denote by $G$ a finite group and by $\\psi(G)$ the sum of element orders in $G$. If $t$ is a positive integer, denote by $C_t$ the cyclic group of order $t$ and write $\\psi(t)=\\psi(C_t)$. In this paper we proved the following Theorem A: Let $G$ be a non-cyclic group of odd order $n=qm$, where $q$ is the smallest prime divisor of $n$ and $(m,q)=1$. Then the following statements hold. (1) If $q=3$, then $\\frac {\\psi(G)}{\\psi(|G|)}\\leq \\frac {85}{301}$, and equality holds if and only if $n=3\\cdot 7\\cdot m_1$ with $(m_1,42)=1$ and $G=(C_7\\rtimes C_3)\\times C_{m_1}$, with $C_7\\rtimes C_3$ non-abelia"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1905.12291","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":"1905.12291","created_at":"2026-05-17T23:44:44.734090+00:00"},{"alias_kind":"arxiv_version","alias_value":"1905.12291v1","created_at":"2026-05-17T23:44:44.734090+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1905.12291","created_at":"2026-05-17T23:44:44.734090+00:00"},{"alias_kind":"pith_short_12","alias_value":"UQD7KM5WF3TP","created_at":"2026-05-18T12:33:30.264802+00:00"},{"alias_kind":"pith_short_16","alias_value":"UQD7KM5WF3TPXQ23","created_at":"2026-05-18T12:33:30.264802+00:00"},{"alias_kind":"pith_short_8","alias_value":"UQD7KM5W","created_at":"2026-05-18T12:33:30.264802+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/UQD7KM5WF3TPXQ23RIRU4GM4O4","json":"https://pith.science/pith/UQD7KM5WF3TPXQ23RIRU4GM4O4.json","graph_json":"https://pith.science/api/pith-number/UQD7KM5WF3TPXQ23RIRU4GM4O4/graph.json","events_json":"https://pith.science/api/pith-number/UQD7KM5WF3TPXQ23RIRU4GM4O4/events.json","paper":"https://pith.science/paper/UQD7KM5W"},"agent_actions":{"view_html":"https://pith.science/pith/UQD7KM5WF3TPXQ23RIRU4GM4O4","download_json":"https://pith.science/pith/UQD7KM5WF3TPXQ23RIRU4GM4O4.json","view_paper":"https://pith.science/paper/UQD7KM5W","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1905.12291&json=true","fetch_graph":"https://pith.science/api/pith-number/UQD7KM5WF3TPXQ23RIRU4GM4O4/graph.json","fetch_events":"https://pith.science/api/pith-number/UQD7KM5WF3TPXQ23RIRU4GM4O4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/UQD7KM5WF3TPXQ23RIRU4GM4O4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/UQD7KM5WF3TPXQ23RIRU4GM4O4/action/storage_attestation","attest_author":"https://pith.science/pith/UQD7KM5WF3TPXQ23RIRU4GM4O4/action/author_attestation","sign_citation":"https://pith.science/pith/UQD7KM5WF3TPXQ23RIRU4GM4O4/action/citation_signature","submit_replication":"https://pith.science/pith/UQD7KM5WF3TPXQ23RIRU4GM4O4/action/replication_record"}},"created_at":"2026-05-17T23:44:44.734090+00:00","updated_at":"2026-05-17T23:44:44.734090+00:00"}