{"paper":{"title":"Zeta-functions of root systems and Poincar\\'e polynomials of Weyl groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Hirofumi Tsumura, Kohji Matsumoto, Yasushi Komori","submitted_at":"2017-07-31T04:24:38Z","abstract_excerpt":"We consider a certain linear combination $S(\\mathbf{s},\\mathbf{y};I;\\Delta)$ of zeta-functions of root systems, where $\\Delta$ is a root system of rank $r$ and $I\\subset\\{1,2,\\ldots,r\\}$. Showing two different expressions of $S(\\mathbf{s},\\mathbf{y};I;\\Delta)$, we find that a certain signed sum of zeta-functions of root systems is equal to a sum involving Bernoulli functions of root systems. This identity gives a non-trivial functional relation among zeta-functions of root systems, if the signed sum does not identically vanish. This is a genralization of the authors' previous result proved in "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.09719","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"}