{"paper":{"title":"Restricted Sum Formula of Alternating Euler Sums","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jianqiang Zhao","submitted_at":"2012-07-23T12:02:43Z","abstract_excerpt":"In this paper we study restricted sum formulas involving alternating Euler sums which are defined by\n  \\zeta(s_1,...,s_{d};\\epsilon_1,...,\\epsilon_d)=\\sum_{n_1>...>n_d\\ge 1}\\frac{\\epsilon_1^{n_1}... \\epsilon_{d}^{n_d}}{n_1^{s_1}... n_d^{s_d}},\nfor all positive integers s_1,...,s_{d} and \\epsilon_1=\\pm 1,..., \\epsilon_{d}=\\pm 1 with (s_1,\\epsilon_1) unequal (1,1). We call w=s_1+...+s_{d} the weight and d the depth. When \\epsilon_j=-1 we say the jth component is alternating. We first consider Euler sums of the following special type:\n  \\xi(2s_1,...,2s_{d})=\\zeta(2s_1,...,2s_{d};(-1)^{s_1},...,(-"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.5366","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"}