{"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"}