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Let be given distinct complex numbers $z_j$ satisfying the conditions $|z_j| = 1, z_j \\not= 1$ for $j=1,..., n$ and for every $z_j$ there exists an $ i$ such that $z_i = \\bar{z_j}. $ Then $$\\inf_{k} \\sum_{j=1}^n z_j^k \\leq - 1. $$ If, moreover, none of the numbers $z_j$ is a root of unity, then $$\\inf_{k} \\sum_{j=1}^n z_j^k \\leq - \\frac {2} {\\pi^3} \\log n. $$ The constant -1 in the former result is the best possible. The above results are special cases of upper bounds for $\\inf_{k} \\sum_{j=1}^n b_jz_j^k$ obtained in this paper."},"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":"1107.5495","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-07-27T14:48:45Z","cross_cats_sorted":[],"title_canon_sha256":"f6f2fa7dcae13642196ea8e253079b34362a00f61a5765848c7ac4bc89e0cb7a","abstract_canon_sha256":"40727446e4a94da4093eed40bf2940c504b09b8b22e7b14775c4d1aa6427c3ef"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:41:29.634484Z","signature_b64":"4PjzMjkWHe0BFsmtj7RAjdJKVYPupfTTK/l7X+sOecR5wo39ir4PraYh869BJpnKlJZwPfN5wk+JK+Bgx3aGCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"628be7527d067c453c7dcf776f69b44490d89b4c5904cf78ed1ae6785c9a7fcd","last_reissued_at":"2026-05-18T03:41:29.633888Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:41:29.633888Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A one-sided power sum inequality","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Frits Beukers, Rob Tijdeman","submitted_at":"2011-07-27T14:48:45Z","abstract_excerpt":"In this note we prove results of the following types. Let be given distinct complex numbers $z_j$ satisfying the conditions $|z_j| = 1, z_j \\not= 1$ for $j=1,..., n$ and for every $z_j$ there exists an $ i$ such that $z_i = \\bar{z_j}. $ Then $$\\inf_{k} \\sum_{j=1}^n z_j^k \\leq - 1. $$ If, moreover, none of the numbers $z_j$ is a root of unity, then $$\\inf_{k} \\sum_{j=1}^n z_j^k \\leq - \\frac {2} {\\pi^3} \\log n. $$ The constant -1 in the former result is the best possible. The above results are special cases of upper bounds for $\\inf_{k} \\sum_{j=1}^n b_jz_j^k$ obtained in this paper."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.5495","kind":"arxiv","version":3},"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":"1107.5495","created_at":"2026-05-18T03:41:29.634001+00:00"},{"alias_kind":"arxiv_version","alias_value":"1107.5495v3","created_at":"2026-05-18T03:41:29.634001+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1107.5495","created_at":"2026-05-18T03:41:29.634001+00:00"},{"alias_kind":"pith_short_12","alias_value":"MKF6OUT5AZ6E","created_at":"2026-05-18T12:26:34.985390+00:00"},{"alias_kind":"pith_short_16","alias_value":"MKF6OUT5AZ6EKPD5","created_at":"2026-05-18T12:26:34.985390+00:00"},{"alias_kind":"pith_short_8","alias_value":"MKF6OUT5","created_at":"2026-05-18T12:26:34.985390+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/MKF6OUT5AZ6EKPD5Z53W62NUIS","json":"https://pith.science/pith/MKF6OUT5AZ6EKPD5Z53W62NUIS.json","graph_json":"https://pith.science/api/pith-number/MKF6OUT5AZ6EKPD5Z53W62NUIS/graph.json","events_json":"https://pith.science/api/pith-number/MKF6OUT5AZ6EKPD5Z53W62NUIS/events.json","paper":"https://pith.science/paper/MKF6OUT5"},"agent_actions":{"view_html":"https://pith.science/pith/MKF6OUT5AZ6EKPD5Z53W62NUIS","download_json":"https://pith.science/pith/MKF6OUT5AZ6EKPD5Z53W62NUIS.json","view_paper":"https://pith.science/paper/MKF6OUT5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1107.5495&json=true","fetch_graph":"https://pith.science/api/pith-number/MKF6OUT5AZ6EKPD5Z53W62NUIS/graph.json","fetch_events":"https://pith.science/api/pith-number/MKF6OUT5AZ6EKPD5Z53W62NUIS/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MKF6OUT5AZ6EKPD5Z53W62NUIS/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MKF6OUT5AZ6EKPD5Z53W62NUIS/action/storage_attestation","attest_author":"https://pith.science/pith/MKF6OUT5AZ6EKPD5Z53W62NUIS/action/author_attestation","sign_citation":"https://pith.science/pith/MKF6OUT5AZ6EKPD5Z53W62NUIS/action/citation_signature","submit_replication":"https://pith.science/pith/MKF6OUT5AZ6EKPD5Z53W62NUIS/action/replication_record"}},"created_at":"2026-05-18T03:41:29.634001+00:00","updated_at":"2026-05-18T03:41:29.634001+00:00"}