{"paper":{"title":"On the absolute convergence of automorphic Dirichlet series","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Ravi Raghunathan","submitted_at":"2021-05-14T16:57:48Z","abstract_excerpt":"Let $F(s)=\\sum_{n=1}^{\\infty}\\frac{a_n}{n^s}$ be a Dirichlet series in the axiomatically defined class ${\\mathfrak A}^{\\#}$ . The class ${\\mathfrak A}^{\\#}$ is known to contain the extended Selberg class ${\\mathcal S}^{\\#}$, as well as all the $L$-functions of automorphic forms on $GL_n/K$, where $K$ is a number field. Let $d$ be the degree of $F(s)$. We show that $\\sum_{n<X}|a_n|=\\Omega(X^{\\frac{1}{2}+\\frac{1}{2d}})$, and hence, that the abscissa of absolute convergence of $\\sigma_a$ of $F(s)$ must satisfy $\\sigma_a\\ge 1/2+1/2d$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2105.06957","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2105.06957/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}