{"paper":{"title":"Linear Independence of Harmonic Numbers over the field of Algebraic Numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Sonika Dhillon, Tapas Chatterjee","submitted_at":"2018-10-23T10:39:42Z","abstract_excerpt":"Let $H_n =\\sum\\limits_{k=1}^n \\frac{1}{k}$ be the $n$-th harmonic number. Euler extended it to complex arguments and defined $H_r$ for any complex number $r$ except for the negative integers. In this paper, we give a new proof of the transcendental nature of $H_r$ for rational $r$. For some special values of $q>1,$ we give an upper bound for the number of linearly independent harmonic numbers $H_{a/q}$ with $ 1 \\leq a \\leq q$ over the field of algebraic numbers. Also, for any finite set of odd primes $J$ with $|J|=n,$ define $$W_J=\\overline{\\mathbb{Q}}-\\text {span of } \\{ H_1, \\ H_{a_{j_i}/q_i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.09763","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"}