{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:YVBNNUHVR2KVM44RRH2KAE5Q6R","short_pith_number":"pith:YVBNNUHV","schema_version":"1.0","canonical_sha256":"c542d6d0f58e9556739189f4a013b0f4578ec37f00ee50aedb42e60f1e21c667","source":{"kind":"arxiv","id":"1105.2603","version":1},"attestation_state":"computed","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 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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 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