{"paper":{"title":"Special values of Dirichlet series and zeta integrals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Aldo Pereira, Eduardo Friedman","submitted_at":"2011-05-13T03:33:07Z","abstract_excerpt":"For $f$ and $g$ polynomials in $p$ variables, we relate the special value at a non-positive integer $s=-N$, obtained by analytic continuation of the Dirichlet series $$ \\zeta(s;f,g)=\\sum_{k_1=0}^\\infty ... \\sum_{k_p=0}^\\infty g(k_1,...,k_p)f(k_1,...,k_p)^{-s}\\ \\,(\\re(s)\\gg0), $$ to special values of zeta integrals $$ Z(s;f,g)=\\int_{x\\in[0,\\infty)^p} g(x)f(x)^{-s}\\,dx \\, \\ (\\re(s)\\gg0).$$ We prove a simple relation between $\\zeta(-N;f,g)$ and $Z(-N;f_a,g_a)$, where for $a\\in\\C ^p,\\ f_a(x)$ is the shifted polynomial $f_a(x)=f(a+x)$.\n  By direct calculation we prove the product rule for zeta inte"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1105.2603","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"}