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We proved that $\\psi(G)\\leq\\frac 7{11}\\psi(C_n)$ and $\\psi(G)<\\frac 1{q-1}\\psi(C_n)$. The first result is best possible, since for each $n=4k$, $k$ odd, there exists a group $G$ of order $n$ satisfying $\\psi(G)=\\frac 7{11}\\psi(C_n)$ and the second result implies that if $G$ is of odd order, then $\\psi(G)<\\frac 12\\psi(C_n)$. Our results improve the inequality $\\psi(G)<\\psi(C_n"},"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":"1610.03669","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-10-12T11:07:28Z","cross_cats_sorted":[],"title_canon_sha256":"49648a2cf806300f9139b04143fa7c2d494ccb62bf5e5ad1ce2f3350485c57bd","abstract_canon_sha256":"139b4d155c0dfe4dcd78da22af05c90029a725db81d34e856db4cc82e5b10e4b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:02:29.619769Z","signature_b64":"B0s9iWmM5+Is76QxHIM1WDDJsxMDNZtkcmN4Ov2aoKKo0zbYVYluao4BTs88o8wZ2cu5cbpgWgTZoVcTRMceAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"59381d6c877297117ae390682b885acdc7609974f7e7670d61300d97653ea385","last_reissued_at":"2026-05-18T01:02:29.619144Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:02:29.619144Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"An exact upper bound for sums of element orders in non-cyclic finite groups","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":"2016-10-12T11:07:28Z","abstract_excerpt":"Denote the sum of element orders in a finite group $G$ by $\\psi(G)$ and let $C_n$ denote the cyclic group of order $n$. Suppose that $G$ is a non-cyclic finite group of order $n$ and $q$ is the least prime divisor of $n$. We proved that $\\psi(G)\\leq\\frac 7{11}\\psi(C_n)$ and $\\psi(G)<\\frac 1{q-1}\\psi(C_n)$. The first result is best possible, since for each $n=4k$, $k$ odd, there exists a group $G$ of order $n$ satisfying $\\psi(G)=\\frac 7{11}\\psi(C_n)$ and the second result implies that if $G$ is of odd order, then $\\psi(G)<\\frac 12\\psi(C_n)$. 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