{"paper":{"title":"Linear independence of values of G-functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"St\\'ephane Fischler (LM-Orsay), Tanguy Rivoal (IF)","submitted_at":"2017-01-31T14:20:33Z","abstract_excerpt":"Given any non-polynomial $G$-function $F(z)=\\sum\\_{k=0}^\\infty A\\_k z^k$ of radius of convergence $R$, we consider the $G$-functions   $F\\_n^{[s]}(z)=\\sum\\_{k=0}^\\infty \\frac{A\\_k}{(k+n)^s}z^k$ for any integers $s\\geq 0$ and $n\\geq 1$. For any fixed algebraic number $\\alpha$ such that $0 \\textless{} \\vert  \\alpha \\vert \\textless{} R$ and any number field $\\mathbb{K}$ containing $\\alpha$ and the $A\\_k$'s, we define $\\Phi\\_{\\alpha, S}$ as the $\\mathbb{K}$-vector space generated by the values $F\\_n^{[s]}(\\alpha)$, $n\\ge 1$ and $0\\leq s\\leq S$. We prove that $u\\_{\\mathbb{K},F}\\log(S)\\leq \\dim\\_{\\m"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.09051","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"}