{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2025:6CNL2KDTL5I6K3QIRRXRLBAMPK","short_pith_number":"pith:6CNL2KDT","schema_version":"1.0","canonical_sha256":"f09abd28735f51e56e088c6f15840c7a891b56a8695b3ee4b133af83cced60b0","source":{"kind":"arxiv","id":"2505.09996","version":2},"attestation_state":"computed","paper":{"title":"Gauss sum with principal multiplicative character","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RT"],"primary_cat":"math.CO","authors_text":"Priya Dhankhar, Sanjay Kumar Singh","submitted_at":"2025-05-15T06:17:56Z","abstract_excerpt":"Let $R$ be a finite ring with unity, $\\psi: R \\to \\mathbb{C}^\\times$ be an additive character of $R$, and \\( \\chi_0 \\) be the principal multiplicative character ($i.e.$, $\\chi_0(x) = 1 \\quad \\text{for all } x \\in R^\\times$), then the Gauss sum is \\[ G(\\chi_0, \\psi) = \\sum_{x \\in R^\\times} \\psi(x). \\] In this paper, we give an explicit formula for a more general form of the Gauss sum $G(\\chi_0, \\psi)$. Interestingly, the formula extends the known formula of classical Ramanujan's sum to the context of finite rings. As an application, we derive the eigenvalues for a more general form of the unita"},"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":"2505.09996","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-05-15T06:17:56Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"c8f3e6b8742fecc50bfb7a6f5a727d7245a7bef360ce11ded1ebe07537c674f6","abstract_canon_sha256":"42289eca32043956d3c6d437dec65894ca55a2e88058287f64c6d28fbf33ee4f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:39:01.355943Z","signature_b64":"LMMdtR9znzwLRI7nalC8rj88LWAs6RHaOPiaxJon3ywwcwxzBUWVaYSinWv78EeHwOWcqs8LXiknzECXAlZXBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f09abd28735f51e56e088c6f15840c7a891b56a8695b3ee4b133af83cced60b0","last_reissued_at":"2026-05-17T23:39:01.355347Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:39:01.355347Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Gauss sum with principal multiplicative character","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RT"],"primary_cat":"math.CO","authors_text":"Priya Dhankhar, Sanjay Kumar Singh","submitted_at":"2025-05-15T06:17:56Z","abstract_excerpt":"Let $R$ be a finite ring with unity, $\\psi: R \\to \\mathbb{C}^\\times$ be an additive character of $R$, and \\( \\chi_0 \\) be the principal multiplicative character ($i.e.$, $\\chi_0(x) = 1 \\quad \\text{for all } x \\in R^\\times$), then the Gauss sum is \\[ G(\\chi_0, \\psi) = \\sum_{x \\in R^\\times} \\psi(x). \\] In this paper, we give an explicit formula for a more general form of the Gauss sum $G(\\chi_0, \\psi)$. Interestingly, the formula extends the known formula of classical Ramanujan's sum to the context of finite rings. As an application, we derive the eigenvalues for a more general form of the unita"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.09996","kind":"arxiv","version":2},"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":"2505.09996","created_at":"2026-05-17T23:39:01.355464+00:00"},{"alias_kind":"arxiv_version","alias_value":"2505.09996v2","created_at":"2026-05-17T23:39:01.355464+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2505.09996","created_at":"2026-05-17T23:39:01.355464+00:00"},{"alias_kind":"pith_short_12","alias_value":"6CNL2KDTL5I6","created_at":"2026-05-18T12:33:37.589309+00:00"},{"alias_kind":"pith_short_16","alias_value":"6CNL2KDTL5I6K3QI","created_at":"2026-05-18T12:33:37.589309+00:00"},{"alias_kind":"pith_short_8","alias_value":"6CNL2KDT","created_at":"2026-05-18T12:33:37.589309+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/6CNL2KDTL5I6K3QIRRXRLBAMPK","json":"https://pith.science/pith/6CNL2KDTL5I6K3QIRRXRLBAMPK.json","graph_json":"https://pith.science/api/pith-number/6CNL2KDTL5I6K3QIRRXRLBAMPK/graph.json","events_json":"https://pith.science/api/pith-number/6CNL2KDTL5I6K3QIRRXRLBAMPK/events.json","paper":"https://pith.science/paper/6CNL2KDT"},"agent_actions":{"view_html":"https://pith.science/pith/6CNL2KDTL5I6K3QIRRXRLBAMPK","download_json":"https://pith.science/pith/6CNL2KDTL5I6K3QIRRXRLBAMPK.json","view_paper":"https://pith.science/paper/6CNL2KDT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2505.09996&json=true","fetch_graph":"https://pith.science/api/pith-number/6CNL2KDTL5I6K3QIRRXRLBAMPK/graph.json","fetch_events":"https://pith.science/api/pith-number/6CNL2KDTL5I6K3QIRRXRLBAMPK/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/6CNL2KDTL5I6K3QIRRXRLBAMPK/action/timestamp_anchor","attest_storage":"https://pith.science/pith/6CNL2KDTL5I6K3QIRRXRLBAMPK/action/storage_attestation","attest_author":"https://pith.science/pith/6CNL2KDTL5I6K3QIRRXRLBAMPK/action/author_attestation","sign_citation":"https://pith.science/pith/6CNL2KDTL5I6K3QIRRXRLBAMPK/action/citation_signature","submit_replication":"https://pith.science/pith/6CNL2KDTL5I6K3QIRRXRLBAMPK/action/replication_record"}},"created_at":"2026-05-17T23:39:01.355464+00:00","updated_at":"2026-05-17T23:39:01.355464+00:00"}